cern - a combination of preliminary electroweak measurements...

EUROPEAN ORGANIZATION FOR NUCLEAR RESEARCH

CERN-EP/2001-021February 28, 2001

A Combination of PreliminaryElectroweak Measurements and

Constraints on the Standard Model

The LEP Collaborations∗ ALEPH, DELPHI, L3, OPAL,the LEP Electroweak Working Group†

and the SLD Heavy Flavour and Electroweak Groups‡

Prepared from Contributions of the LEP and SLD experimentsto the 2000 Summer conferences.

Abstract

This note presents a combination of published and preliminary electroweak results from thefour LEP collaborations and the SLD collaboration which were prepared for the 2000 summerconferences. Averages from Z resonance results are derived for hadronic and leptonic cross sections,the leptonic forward-backward asymmetries, the τ polarisation asymmetries, the bb and cc partialwidths and forward-backward asymmetries and the qq charge asymmetry. Above the Z resonance,averages are derived for di–fermion cross sections and asymmetries, W–pair, Z–pair and single–Wproduction cross section, electroweak gauge boson couplings and W mass and decay branchingratios. The major changes with respect to results presented in summer 1999 are final Z lineshaperesults from LEP, updates to the W mass and gauge-boson couplings from LEP, and ALR fromSLD. The results are compared with precise electroweak measurements from other experiments.The parameters of the Standard Model are evaluated, first using the combined LEP electroweakmeasurements, and then using the full set of electroweak results.

∗The LEP Collaborations each take responsibility for the preliminary results of their own.†The members of the LEP Electroweak Working Group who contributed significantly to this note are : D. Ab-

baneo, J. Alcaraz, P. Antilogus, S. Arcelli, P. Bambade, E. Barberio, G. Bella, A. Blondel, S. Blyth, D. Bourilkov,R. Chierici, R. Clare, P. de Jong, G. Duckeck, A. Ealet, M. Elsing, F. Fiedler, P. Garcia-Abia, A. Gurtu, M.W. Grunewald,J.B. Hansen, R. Hawkings, J. Holt, S. Jezequel, R.W.L. Jones, N. Kjaer, M. Kobel, E. Lancon, W. Lohmann, C. Mar-iotti, M. Martinez, F. Matorras, C. Matteuzzi, S. Mele, E. Migliore, M.N. Minard, K. Monig, A. Olshevski, C. Parkes,U. Parzefall, Ch. Paus, M. Pepe-Altarelli, B. Pietrzyk, G. Quast, P. Renton, H. Rick, S. Riemann, J.M. Roney, K. Sachs,C. Sbarra, A. Schmidt-Kaerst, S. Spagnolo, A. Straessner, R. Strohmer D. Strom, R. Tenchini, F. Terranova, F. Teubert,M.A. Thomson, E. Tournefier, M. Verzocchi, H. Voss, C.P. Ward, St. Wynhoff.

‡T. Abe, N. de Groot, M. Iwasaki, P.C. Rowson, D. Su, M. Swartz.

1 Introduction

The four LEP experiments and SLD have previously presented [1] parameters derived from the Zresonance using published and preliminary results based on data recorded until the end of 1995 for theLEP experiments and 1998 for SLD. Since 1996 LEP has run at energies above the W-pair productionthreshold. In 1999 the delivered luminosity was significantly higher than in previous years, and thusthe knowledge of the properties of the W boson has been significantly improved. To allow a quickassessment, a box highlighting the updates is given at the beginning of each section.

LEP-I (1990-1995) Z-pole measurements are the hadronic and leptonic cross sections, the leptonicforward-backward asymmetries, the τ polarisation asymmetries, the bb and cc partial widths andforward-backward asymmetries and the qq charge asymmetry. The measurements of the left-rightcross section asymmetry, the bb and cc partial widths and left-right-forward-backward asymmetriesfor b and c quarks from SLD are treated consistently with the LEP data. Many technical aspects oftheir combination are described in References 2, 3 and references therein.

The LEP-II (1996-2000) measurements are di–fermion cross sections and asymmetries; W–pair, Z–pair and single–W production cross sections, electroweak gauge boson couplings. W boson properties,like mass, width and decay branching ratios are also measured.

Several measurements included in the combinations are still preliminary.

This note is organised as follows:

Section 2 Z Line Shape and Leptonic Forward-Backward Asymmetries;

Section 3 τ Polarisation;

Section 4 ALR Measurement at SLD;

Section 5 Heavy Flavour Analyses;

Section 6 Inclusive Hadronic Charge Asymmetry;

Section 7 ff Production at Energies above the Z;

Section 8 W Boson Properties, including mW, Branching Ratios, W–pair Production Cross Section;

Section 9 Single–W Production Cross Section;

Section 10 ZZ Production Cross Section;

Section 11 Electroweak Gauge Boson Couplings;

Section 12 Interpretation of the Results, Including the Combination of Results from LEP, SLD,Neutrino Interaction Experiments and from CDF and DØ;

Section 13 Prospects for the Future.

2

2 Results from the Z Peak Data

Updates with respect to last summer:All experiments have updated their results, and all are now final. Recent theoretical developmentshave been included in the results and fits.

2.1 Z Lineshape and Lepton Forward-Backward Asymmetries

The results presented here are based on the full LEP-I data set. This includes the data taken duringthe energy scans in 1990 and 1991 in the range1 |√s−mZ| < 3 GeV, the data collected at the Z peakin 1992 and 1994 and the precise energy scans in 1993 and 1995 (|√s −mZ| < 1.8 GeV). The totalevent statistics are given in Table 1. Details of the individual analyses can be found in References 4–7.

qqyear A D L O all

’90/91 433 357 416 454 1660’92 633 697 678 733 2741’93 630 682 646 649 2607’94 1640 1310 1359 1601 5910’95 735 659 526 659 2579

total 4071 3705 3625 4096 15497

`+`−

year A D L O all’90/91 53 36 39 58 186

’92 77 70 59 88 294’93 78 75 64 79 296’94 202 137 127 191 657’95 90 66 54 81 291

total 500 384 343 497 1724

Table 1: The qq and `+`− event statistics, in units of 103, used for the analysis of the Z line shapeand lepton forward-backward asymmetries by the experiments ALEPH (A), DELPHI (D), L3 (L) andOPAL (O).

For the averaging of results the LEP experiments provide a standard set of 9 parameters describingthe information contained in hadronic and leptonic cross sections and leptonic forward-backward asym-metries. These parameters are convenient for fitting and averaging since they have small correlations.They are:

• The mass and total width of the Z boson, where the definition is based on the Breit-Wignerdenominator (s−m2

Z + isΓZ/mZ) with s-dependent width [8].

• The hadronic pole cross section of Z exchange:

σ0h ≡

12πm2

Z

ΓeeΓhad

Γ2Z

. (1)

Here Γee and Γhad are the partial widths of the Z for decays into electrons and hadrons.

• The ratios:Re ≡ Γhad/Γee, Rµ ≡ Γhad/Γµµ and Rτ ≡ Γhad/Γττ . (2)

Here Γµµ and Γττ are the partial widths of the Z for the decays Z → µ+µ− and Z → τ+τ−. Dueto the mass of the τ lepton, a difference of 0.2% is expected between the values for Re and Rµ,and the value for Rτ , even under the assumption of lepton universality [9].

1In this note h = c = 1.

3

• The pole asymmetries, A0, eFB, A0, µ

FB and A0, τFB , for the processes e+e− → e+e−, e+e− → µ+µ− and

e+e− → τ+τ−. In terms of the real parts of the effective vector and axial-vector neutral currentcouplings of fermions, gV f and gAf , the pole asymmetries are expressed as

A0, fFB ≡

34AeAf (3)

with

Af ≡ 2gV fgAf

g2V f + g2

Af

= 2gV f/gAf

1 + (gV f/gAf)2. (4)

The imaginary parts of the vector and axial-vector coupling constants as well as real and imaginaryparts of the photon vacuum polarisation are taken into account explicitly in the fitting formulae andare fixed to their Standard Model values. The fitting procedure takes into account the effects of initial-state radiation [8] to O(α3) [10–12], as well as the t-channel and the s-t interference contributions inthe case of e+e− final states.

The set of 9 parameters does not describe hadron and lepton-pair production completely, becauseit does not include the interference of the s-channel Z exchange with the s-channel γ exchange. Forthe results presented in this section and used in the rest of the note, the γ-exchange contributionsand the hadronic γZ interference terms are fixed to their Standard Model values. The leptonic γZinterference terms are expressed in terms of the effective couplings.

The four sets of 9 parameters provided by the LEP experiments are presented in Table 2. For per-forming the average over these four sets of nine parameters, the overall covariance matrix is constructedfrom the covariance matrices of the individual LEP experiments and common systematic errors [2].The common systematic errors include theoretical errors as well as errors arising from the uncertaintyin the LEP beam energy. The beam energy uncertainty contributes an uncertainty of ±1.7 MeV tomZ and ±1.2 MeV to ΓZ. In addition, the uncertainty in the centre-of-mass energy spread of about±1 MeV contributes ±0.2 MeV to ΓZ. The theoretical error on calculations of the small-angle Bhabhacross section is ±0.054 % [13] for OPAL and ±0.061 % [14] for all other experiments, and results inthe largest common systematic uncertainty on σ0

h. QED radiation, dominated by photon radiationfrom the initial state electrons, contributes a common uncertainty of ±0.02 % on σ0

h, of ±0.3 MeVon mZ and of ±0.2 MeV on ΓZ. The contribution of t-channel diagrams and the s-t interference inZ → e+e− leads to an additional theoretical uncertainty estimated to be ±0.24 on Re and ±0.0014 onA0, e

FB, which are fully anti–correlated. Uncertainties from the model-independent parameterisation ofthe energy dependence of the cross section are almost negligible, if the definitions of Reference [15]are applied. Through unavoidable Standard Model remnants, dominated by the need to fix the γ-Zinterference contribution in the qq channel, there is some small dependence of ±0.2 MeV of mZ on theHiggs mass, mH (in the range 100 GeV to 1000 GeV) and the value of the electromagnetic couplingconstant. Such “parametric” errors are negligible for the other pseudo-observables. The combinedparameter set and its correlation matrix are given in Table 3.

If lepton universality is assumed, the set of 9 parameters is reduced to a set of 5 parameters.R` is defined as R` ≡ Γhad/Γ``, where Γ`` refers to the partial Z width for the decay into a pair ofmassless charged leptons. The data of each of the four LEP experiments are consistent with leptonuniversality (the difference in χ2 over the difference in d.o.f. with and without the assumption oflepton universality is 3/4, 6/4, 5/4 and 3/4 for ALEPH, DELPHI, L3 and OPAL, respectively). Thelower part of Table 3 gives the combined result and the corresponding correlation matrix. Figure 1shows, for each lepton species and for the combination assuming lepton universality, the resulting 68%probability contours in the R`-A

0, `FB plane. Good agreement is observed.

4

correlationsmZ ΓZ σ0

h Re Rµ Rτ A0, eFB A0, µ

FB A0, τFB

χ2/Ndf = 169/176 ALEPHmZ [GeV] 91.1891 ± 0.0031 1.00ΓZ [GeV] 2.4959 ± 0.0043 .038 1.00σ0

h [nb] 41.558 ± 0.057 −.091−.383 1.00Re 20.690 ± 0.075 .102 .004 .134 1.00Rµ 20.801 ± 0.056 −.003 .012 .167 .083 1.00Rτ 20.708 ± 0.062 −.003 .004 .152 .067 .093 1.00A0, e

FB 0.0184 ± 0.0034 −.047 .000−.003−.388 .000 .000 1.00A0, µ

FB 0.0172 ± 0.0024 .072 .002 .002 .019 .013 .000−.008 1.00A0, τ

FB 0.0170 ± 0.0028 .061 .002 .002 .017 .000 .011−.007 .016 1.00

χ2/Ndf = 177/168 DELPHImZ [GeV] 91.1864 ± 0.0028 1.00ΓZ [GeV] 2.4876 ± 0.0041 .047 1.00σ0

h [nb] 41.578 ± 0.069 −.070−.270 1.00Re 20.88 ± 0.12 .063 .000 .120 1.00Rµ 20.650 ± 0.076 −.003−.007 .191 .054 1.00Rτ 20.84 ± 0.13 .001−.001 .113 .033 .051 1.00A0, e

FB 0.0171 ± 0.0049 .057 .001−.006−.106 .000−.001 1.00A0, µ

FB 0.0165 ± 0.0025 .064 .006−.002 .025 .008 .000−.016 1.00A0, τ

FB 0.0241 ± 0.0037 .043 .003−.002 .015 .000 .012−.015 .014 1.00

χ2/Ndf = 158/166 L3mZ [GeV] 91.1897 ± 0.0030 1.00ΓZ [GeV] 2.5025 ± 0.0041 .065 1.00σ0

h [nb] 41.535 ± 0.054 .009−.343 1.00Re 20.815 ± 0.089 .108−.007 .075 1.00Rµ 20.861 ± 0.097 −.001 .002 .077 .030 1.00Rτ 20.79 ± 0.13 .002 .005 .053 .024 .020 1.00A0, e

FB 0.0107 ± 0.0058 −.045 .055−.006−.146−.001−.003 1.00A0, µ

FB 0.0188 ± 0.0033 .052 .004 .005 .017 .005 .000 .011 1.00A0, τ

FB 0.0260 ± 0.0047 .034 .004 .003 .012 .000 .007−.008 .006 1.00

χ2/Ndf = 155/194 OPALmZ [GeV] 91.1858 ± 0.0030 1.00ΓZ [GeV] 2.4948 ± 0.0041 .049 1.00σ0

h [nb] 41.501 ± 0.055 .031−.352 1.00Re 20.901 ± 0.084 .108 .011 .155 1.00Rµ 20.811 ± 0.058 .001 .020 .222 .093 1.00Rτ 20.832 ± 0.091 .001 .013 .137 .039 .051 1.00A0, e

FB 0.0089 ± 0.0045 −.053−.005 .011−.222−.001 .005 1.00A0, µ

FB 0.0159 ± 0.0023 .077−.002 .011 .031 .018 .004−.012 1.00A0, τ

FB 0.0145 ± 0.0030 .059−.003 .003 .015−.010 .007−.010 .013 1.00

Table 2: Line Shape and asymmetry parameters from fits to the data of the four LEP experimentsand their correlation coefficients.

For completeness the partial decay widths of the Z boson are listed in Table 4, although they aremore correlated than the ratios given in Table 3. The leptonic pole cross-section, σ0

` , defined as

σ0` ≡

12πm2

Z

Γ2``

Γ2Z

,

in analogy to σ0h, is shown in the last line of the Table. Because QCD final state corrections appear

5

quadratically in the denominator via ΓZ, σ0` has a higher sensitivity to αs than σ0

h or R`, where thedependence on QCD corrections is only linear.

without lepton universality correlationsχ2/Ndf = 32.6/27 mZ ΓZ σ0

h Re Rµ Rτ A0, eFB A0, µ

FB A0, τFB

mZ [GeV] 91.1876± 0.0021 1.00ΓZ [GeV] 2.4952 ± 0.0023 −.024 1.00σ0

h [nb] 41.541 ± 0.037 −.044−.297 1.00Re 20.804 ± 0.050 .078−.011 .105 1.00Rµ 20.785 ± 0.033 .000 .008 .131 .069 1.00Rτ 20.764 ± 0.045 .002 .006 .092 .046 .069 1.00A0, e

FB 0.0145 ± 0.0025 −.014 .007 .001−.371 .001 .003 1.00A0, µ

FB 0.0169 ± 0.0013 .046 .002 .003 .020 .012 .001−.024 1.00A0, τ

FB 0.0188 ± 0.0017 .035 .001 .002 .013−.003 .009−.020 .046 1.00

with lepton universalityχ2/Ndf = 36.5/31 mZ ΓZ σ0

h R` A0, `FB

mZ [GeV] 91.1875± 0.0021 1.00ΓZ [GeV] 2.4952 ± 0.0023 −.023 1.00σ0

h [nb] 41.540 ± 0.037 −.045−.297 1.00R` 20.767 ± 0.025 .033 .004 .183 1.00A0, `

FB 0.0171 ± 0.0010 .055 .003 .006−.056 1.00

Table 3: Average line shape and asymmetry parameters from the data of the four LEP experiments,without and with the assumption of lepton universality.

without lepton universality correlationsΓhad [MeV] 1745.8 ±2.7 1.00Γee [MeV] 83.92±0.12 −0.29 1.00Γµµ [MeV] 83.99±0.18 0.66−0.20 1.00Γττ [MeV] 84.08±0.22 0.54−0.17 0.39 1.00

with lepton universalityΓinv [MeV] 499.0 ±1.5 1.00Γhad [MeV] 1744.4 ±2.0 −0.29 1.00Γ`` [MeV] 83.984±0.086 0.49 0.39 1.00Γinv/Γ`` 5.942 ±0.016σ0

` [nb] 2.0003±0.0027

Table 4: Partial decay widths of the Z boson, derived from the results of the 9-parameter averages inTable 3. In the case of lepton universality, Γ`` refers to the partial Z width for the decay into a pairof massless charged leptons.

6

0.01

0.014

0.018

0.022

20.6 20.7 20.8 20.9

Rl

A0,

l

fb68% CL

l+l−

e+e−

µ+µ−

τ+τ−

αs

mt

mH

Figure 1: Contours of 68% probability in the R`-A0, `FB plane. For better comparison the results for

the τ lepton are corrected to correspond to the massless case. The Standard Model prediction formZ = 91.1875 GeV, mt = 174.3 GeV, mH = 300 GeV, and αs(m2

Z) = 0.119 is also shown. The lineswith arrows correspond to the variation of the Standard Model prediction when mt, mH and αs(m2

Z)are varied in the intervals mt = 174.3 ± 5.1 GeV, mH = 300+700

−187 GeV, and αs(m2Z) = 0.119 ± 0.002,

respectively. The arrows point in the direction of increasing values of mt, mH and αs.

7

3 The τ Polarisation

Updates with respect to last summer:DELPHI have finalised their results.OPAL have updated their results.

The longitudinal τ polarisation Pτ of τ pairs produced in Z decays is defined as

Pτ ≡ σR − σL

σR + σL, (5)

where σR and σL are the τ -pair cross sections for the production of a right-handed and left-handedτ−, respectively. The distribution of Pτ as a function of the polar scattering angle θ between the e−

and the τ−, at√s = mZ, is given by

Pτ (cos θ) = −Aτ (1 + cos2 θ) + 2Ae cos θ1 + cos2 θ + 2AτAe cos θ

, (6)

with Ae and Aτ as defined in Equation (4). Equation (6) is valid for pure Z exchange. The effects ofγ exchange, γ-Z interference and electromagnetic radiative corrections in the initial and final statesare taken into account in the experimental analyses. In particular, these corrections account for the√s dependence of the τ polarisation, which is important because the off-peak data are included in the

event samples for all experiments. When averaged over all production angles Pτ is a measurement ofAτ . As a function of cos θ, Pτ (cos θ) provides nearly independent determinations of both Aτ and Ae,thus allowing a test of the universality of the couplings of the Z to e and τ .

Each experiment makes separate Pτ measurements using the five τ decay modes eνν, µνν, πν, ρνand a1ν [16–19]. The ρν and πν are the most sensitive channels, contributing weights of about 40%each in the average. DELPHI and L3 have also used an inclusive hadronic analysis. The combinationis made using the results from each experiment already averaged over the τ decay modes.

3.1 Results

Tables 5 and 6 show the most recent results for Aτ andAe obtained by the four LEP collaborations [16–19] and their combination. Common systematic errors arise from uncertainties in the decay radiationin the πν and ρν channels, and in the modelling of the a1 decays [20]. These errors need furtherinvestigation and might need to be taken into account for the final results (see Reference 18). Forthe current combination the systematic errors on Aτ and Ae are treated as uncorrelated between theexperiments. The statistical correlation between the extracted values of Aτ and Ae is small (≤ 5%),and is neglected.

The average values for Aτ and Ae:

Aτ = 0.1439 ± 0.0042 (7)Ae = 0.1498 ± 0.0048 , (8)

are compatible, in agreement with lepton universality. Assuming e-τ universality, the values for Aτ

and Ae can be combined. This combination is performed neglecting any possible common systematicerror between Aτ and Ae within a given experiment, as these errors are also estimated to be small.The combined result of Aτ and Ae is:

A` = 0.1464 ± 0.0032 . (9)

8

Experiment Aτ

ALEPH (90 - 95), prel. 0.1452 ± 0.0052 ± 0.0032DELPHI (90 - 95), final 0.1359 ± 0.0079 ± 0.0055L3 (90 - 95), final 0.1476 ± 0.0088 ± 0.0062OPAL (90 - 95), prel. 0.1456 ± 0.0075 ± 0.0057

LEP Average 0.1439 ± 0.0042

Table 5: LEP results for Aτ . The χ2/d.o.f. for the average is 0.9/3. The first error is statisticaland the second systematic. In the LEP average, statistical and systematic errors are combined inquadrature. The systematic component of the error is ±0.0023.

Experiment Ae

ALEPH (90 - 95), prel. 0.1505 ± 0.0069 ± 0.0010DELPHI (90 - 95), final 0.1382 ± 0.0116 ± 0.0005L3 (90 - 95), final 0.1678 ± 0.0127 ± 0.0030OPAL (90 - 95), prel. 0.1456 ± 0.0103 ± 0.0034

LEP Average 0.1498 ± 0.0048

Table 6: LEP results for Ae. The χ2/d.o.f. for the average is 3.1/3. The first error is statisticaland the second systematic. In the LEP average, statistical and systematic errors are combined inquadrature. The systematic component of the error is ±0.0010.

9

4 Measurement of ALR at SLC

Updates with respect to last summer:SLD have final results for ALR and the leptonic left-right forward-backward asymmetries.

The measurement of the left-right cross section asymmetry (ALR) by SLD [21] at the SLC providesa systematically precise, statistics-dominated determination of the coupling Ae, and is presently themost precise single measurement, with the smallest systematic error, of this quantity. In principlethe analysis is straightforward: one counts the numbers of Z bosons produced by left and rightlongitudinally polarised electrons, forms an asymmetry, and then divides by the luminosity-weightede− beam polarisation magnitude (the e+ beam is not polarised):

ALR =NL −NR

NL +NR

1Pe. (10)

Since the advent of high polarisation “strained lattice” GaAs photocathodes (1994), the average elec-tron polarisation at the interaction point has been in the range 73% to 77%. The method requires nodetailed final state event identification (e+e− final state events are removed, as are non-Z backgrounds)and is insensitive to all acceptance and efficiency effects. The small total systematic error of 0.64% isdominated by the 0.50% systematic error in the determination of the e− polarisation. The statisticalerror on ALR is about 1.3%.

The precision Compton polarimeter detects beam electrons that have been scattered by photonsfrom a circularly polarised laser. Two additional polarimeters that are sensitive to the Compton-scattered photons and which are operated in the absence of positron beam, have verified the precisionpolarimeter result and are used to set a calibration uncertainty of 0.4%. In 1998, a dedicated ex-periment was performed in order to directly test the expectation that accidental polarisation of thepositron beam was negligible; the e+ polarisation was found to be consistent with zero (−0.02±0.07)%.

The ALR analysis includes several very small corrections. The polarimeter result is corrected forhigher order QED and accelerator related effects, a total of (−0.22 ± 0.15)% for 1997/98 data. Theevent asymmetry is corrected for backgrounds and accelerator asymmetries, a total of (+0.15±0.07)%,for 1997/98 data.

The translation of the ALR result to a “pole” value is a (−2.5±0.4)% effect, where the uncertaintyarises from the precision of the centre-of-mass energy determination. This small error due to thebeam energy measurement is slightly larger than seen previously (it was closer to 0.3%) and reflectsthe results of a scan of the Z peak used to calibrate the energy spectrometers to mZ from LEP data,which was performed for the first time during the most recent SLC run. The pole value, A0

LR, isequivalent to a measurement of Ae.

The 2000 result is included in a running average of all of the SLD ALR measurements (1992, 1993,1994/1995, 1996, 1997 and 1998). This updated result for A0

LR (Ae) is 0.1514±0.0022. In addition, theleft-right forward-backward asymmetries for leptonic final states are measured [22]. From these, theparameters Ae, Aµ andAτ can be determined. The results areAe = 0.1544±0.0060, Aµ = 0.142±0.015and Aτ = 0.136± 0.015. The lepton-based result for Ae can be combined with the A0

LR result to yieldAe = 0.1516 ± 0.0021, including small correlations in the systematic errors. The correlation of thismeasurement with Aµ and Aτ is indicated in Table 7.

Assuming lepton universality, the ALR result and the results on the leptonic left-right forward-backward asymmetries can be combined, while accounting for small correlated systematic errors,

10

yieldingA` = 0.1513 ± 0.0021. (11)

Ae Aµ Aτ

Ae 1.000Aµ 0.038 1.000Aτ 0.033 0.007 1.000

Table 7: Correlation coefficients between Ae, Aµ and Aτ

11

5 Results from b and c Quarks

Updates with respect to last summer:DELPHI has presented new measurements of Abb

FB and AccFB.

SLD has presented updated measurements of Rb, Rc, Ab with leptons and vertex charge and Ac withleptons and D-mesons.ALEPH has presented a new measurement of BR(b → `) and BR(b → c → ¯) and L3 has publishedtheir measurement of Rb and BR(b → `).

The relevant quantities in the heavy quark sector at LEP/SLD which are currently determined by thecombination procedure are:

• The ratios of the b and c quark partial widths of the Z to its total hadronic partial width:R0

b ≡ Γbb/Γhad and R0c ≡ Γcc/Γhad.

• The forward-backward asymmetries, AbbFB and Acc

FB.

• The final state coupling parameters Ab, Ac obtained from the left-right-forward-backward asym-metry at SLD.

• The semileptonic branching ratios, BR(b → `), BR(b → c → ¯) and BR(c → `), and the averagetime-integrated B0B0 mixing parameter, χ. These are often determined at the same time orwith similar methods as the asymmetries. Including them in the combination greatly reducesthe errors. For example the measurements of χ act as an effective measurement of the chargetagging efficiency, so that all errors coming from the mixture of different lepton sources in bbevents cancel in the asymmetries.

• The probability that a c quark produces a D+, Ds, D∗+ meson2 or a charmed baryon. The prob-ability that a c quark fragments into a D0 is calculated from the constraint that the probabilitiesfor the weakly decaying charmed hadrons add up to one.

A full description of the averaging procedure is published in [3]; the main motivations for the procedureare outlined here. Several analyses measure more than one parameter simultaneously, for example theasymmetry measurements with leptons or D mesons. Some of the measurements of electroweak pa-rameters depend explicitly on the values of other parameters, for example Rb depends on Rc. Thecommon tagging and analysis techniques lead to common sources of systematic uncertainty, in partic-ular for the double-tag measurements of Rb. The starting point for the combination is to ensure thatall the analyses use a common set of assumptions for input parameters which give rise to systematicuncertainties. The input parameters are updated and extended [1, 23] to accommodate new analysesand more recent measurements. The correlations and interdependences of the input measurements arethen taken into account in a χ2 minimisation which results in the combined electroweak parametersand their correlation matrix.

In a first fit the asymmetry measurements on peak, above peak and below peak are corrected tothree common centre-of-mass energies and are then combined at each energy point. The results of thisfit, including the SLD results, are given in Appendix A. The dependence of the average asymmetrieson centre-of-mass energy agrees with the prediction of the Standard Model. A second fit is made toderive the pole asymmetries A0, q

FB from the measured quark asymmetries, in which all the off-peak2Actually the product P(c → D∗+) × BR(D∗+ → π+D0) is fitted because this quantity is needed and measured by

the LEP experiments.

12

asymmetry measurements are corrected to the peak energy before combining. This fit determinesa total of 14 parameters: the two partial widths, two LEP asymmetries, two coupling parametersfrom SLD, three semileptonic branching ratios, the average mixing parameter and the probabilitiesfor c quark to fragment into a D+, a Ds, a D∗+, or a charmed baryon. If the SLD measurements areexcluded from the fit there are 12 parameters to be determined.

5.1 Summary of Measurements and Averaging Procedure

All measurements are presented by the LEP and SLD collaborations in a consistent manner for thepurpose of combination. The tables prepared by the experiments include a detailed breakdown of thesystematic error of each measurement and its dependence on other electroweak parameters. Wherenecessary, the experiments apply small corrections to their results in order to use agreed values andranges for the input parameters to calculate systematic errors. The measurements, corrected wherenecessary, are summarised in Appendix A in Tables 44–63, where the statistical and systematic errorsare quoted separately. The correlated systematic entries are from physics sources shared with oneor more other results in the table and are derived from the full breakdown of common systematicuncertainties. The uncorrelated systematic entries come from the remaining sources.

5.1.1 Averaging Procedure

A χ2 minimisation procedure is used to derive the values of the heavy-flavour electroweak parametersas published in Reference 3. The full statistical and systematic covariance matrix for all measurementsis calculated. This correlation matrix takes into account correlations between different measurementsof one experiment and between different experiments. The explicit dependence of each measurementon the other parameters is also accounted for.

Since c-quark events form the main background in the Rb analyses, in the lifetime Rb analyses,the value of Rb depends on the value of Rc. If Rb and Rc are measured in the same analysis, this isreflected in the correlation matrix for the results. However the analyses do not determine Rb and Rc

simultaneously but instead measure Rb for an assumed value of Rc. In this case the dependence isparameterised as

Rb = Rmeasb + a(Rc)

(Rc −Rusedc )

Rc. (12)

In this expression, Rmeasb is the result of the analysis assuming a value of Rc = Rused

c . The valuesof Rused

c and the coefficients a(Rc) are given in Table 44 where appropriate. The dependence of allother measurements on other electroweak parameters is treated in the same way, with coefficients a(x)describing the dependence on parameter x.

5.1.2 Partial Width Measurements

The measurements of Rb and Rc fall into two categories. In the first, called a single-tag measurement,a method to select b or c events is devised, and the number of tagged events is counted. This numbermust then be corrected for backgrounds from other flavours and for the tagging efficiency to calculatethe true fraction of hadronic Z decays of that flavour. The dominant systematic errors come fromunderstanding the branching ratios and detection efficiencies which give the overall tagging efficiency.

13

For the second technique, called a double-tag measurement, each event is divided into two hemispheres.With Nt being the number of tagged hemispheres, Ntt the number of events with both hemispherestagged and Nhad the total number of hadronic Z decays one has

Nt

2Nhad= εbRb + εcRc + εuds(1−Rb −Rc), (13)

Ntt

Nhad= Cbε

2bRb + Ccε

2cRc + Cudsε

2uds(1−Rb −Rc), (14)

where εb, εc and εuds are the tagging efficiencies per hemisphere for b, c and light-quark events, andCq 6= 1 accounts for the fact that the tagging efficiencies between the hemispheres may be correlated.In the case of Rb one has εb � εc � εuds, Cb ≈ 1. The correlations for the other flavours can beneglected. These equations can be solved to give Rb and εb. Neglecting the c and uds backgroundsand the correlations they are approximately given by

εb ≈ 2Ntt/Nt, (15)Rb ≈ N2

t /(4NttNhad). (16)

The double-tagging method has the advantage that the b tagging efficiency is derived from the data,reducing the systematic error. The residual background of other flavours in the sample, and theevaluation of the correlation between the tagging efficiencies in the two hemispheres of the event arethe main sources of systematic uncertainty in such an analysis.

This method can be enhanced by including more tags. All additional efficiencies can be determinedfrom the data, reducing the statistical uncertainties without adding new systematic uncertainties.

Small corrections must be applied to the results to obtain the partial width ratios R0b and R0

c fromthe cross section ratios Rb and Rc. These corrections depend slightly on the invariant mass cutoff ofthe simulations used by the experiments, so that they are applied by the collaborations before thecombination.

The partial width measurements included are:

• Lifetime (and lepton) double tag measurements for Rb from ALEPH [24], DELPHI [25], L3[26], OPAL [27] and SLD [28]. These are the most precise determinations of Rb. Since theycompletely dominate the combined result, no other Rb measurements are used at present. Thebasic features of the double-tag technique are discussed above. In the ALEPH, DELPHI, OPALand SLD measurements the charm rejection is enhanced by using the invariant mass information.DELPHI, OPAL and SLD also add kinematic information from the particles at the secondaryvertex. The ALEPH and DELPHI measurements make use of several different tags; this improvesthe statistical accuracy and reduces the systematic errors due to hemisphere correlations andcharm contamination, compared with the simple single/double tag.

• Analyses with D/D∗± mesons to measure Rc from ALEPH, DELPHI and OPAL. All measure-ments are constructed in such a way that no assumptions on the energy dependence of charmfragmentation are necessary. The available measurements can be divided into four groups:

– inclusive/exclusive double tag (ALEPH [29], DELPHI [30, 31], OPAL [32]): In a first stepD∗± mesons are reconstructed in several decay channels and their production rate is mea-sured, which depends on the product Rc × P(c → D∗+)× BR(D∗+ → π+D0). This sampleof cc (and bb) events is then used to measure P(c → D∗+)×BR(D∗+ → π+D0) using a slowpion tag in the opposite hemisphere. In the ALEPH measurement Rc is unfolded internallyin the analysis so that no explicit P(c → D∗+)× BR(D∗+ → π+D0) is available.

14

– exclusive double tag (ALEPH [29]): This analysis uses exclusively reconstructed D∗+, D0

and D+ mesons in different decay channels. It has lower statistics but better purity thanthe inclusive analyses.

– reconstruction of all weakly decaying charmed states (ALEPH [33], DELPHI [31], OPAL[34]): These analyses make the assumption that the production rates of D0, D+, Ds and Λc

in cc events add up to one with small corrections due to unmeasured charms strange baryons.This is a single tag measurement, relying only on knowing the decay branching ratios of thecharm hadrons. These analyses are also used to measure the c hadron production ratioswhich are needed for the Rb analyses.

• A lifetime plus mass double tag from SLD to measureRc [28]. This analysis uses the same taggingalgorithm as the SLD Rb analysis, but with the neural net tuned to tag charm. Although thecharm tag has a purity of about 84%, most of the background is from b which can be measuredwith high precision from the b/c mixed tag rate.

• A measurement of Rc using single leptons assuming BR(c → `) from ALEPH [29].

To avoid effects from non linearities in the fit, for the inclusive/exclusive single/double tag and forthe charm-counting analyses, the products RcP(c → D∗+)× BR(D∗+ → π+D0), RcfD0, RcfD+ , RcfDs

and RcfΛcthat are actually measured in the analyses are directly used as inputs to the fit. Themeasurements of the production rates of weakly decaying charmed hadrons, especially RcfDs andRcfΛc have a substantial error due to the branching ratio of the decay mode used. Since this erroris a relative one there is a potential bias towards lower measurements. To avoid this bias, for theproduction rates of weakly decaying charmed hadrons the logarithm of the production rates insteadof the rates themselves are input to the fit. For RcfD0 and RcfD+ the difference between the resultsusing the logarithm or the value itself is negligible. For RcfDs and RcfΛc the difference in the Rc-resultis about one tenth of a standard deviation.

5.1.3 Asymmetry Measurements

All b and c asymmetries given by the experiments are corrected to full acceptance.

The QCD corrections to the forward-backward asymmetries depend strongly on the experimentalanalyses. For this reason the numbers given by the collaborations are also corrected for QCD effects.A detailed description of the procedure can be found in [35] with updates reported in [1]

For the 12- and 14-parameter fits described above, the LEP peak and off-peak asymmetries arecorrected to

√s = 91.26 GeV using the predicted dependence from ZFITTER [36]. The slope of the

asymmetry around mZ depends only on the axial coupling and the charge of the initial and final statefermions and is thus independent of the value of the asymmetry itself.

After calculating the overall averages, the quark pole asymmetries, A0, qFB , are derived by applying

the corrections described below. To relate the pole asymmetries to the measured ones a few correctionsthat are summarised in Table 8 have to be applied. These corrections are due to the energy shift from91.26 GeV to mZ, initial state radiation, γ exchange and γ-Z interference. A very small correctiondue to the nonzero value of the b quark mass is included in the correction called γ-Z interference. Allcorrections are calculated using ZFITTER.

The SLD left-right-forward-backward asymmetries are also corrected for all radiative effects andare directly presented in terms of Ab and Ac.

15

Source δAbFB δAc

FB√s = mZ −0.0013 −0.0034

QED corrections +0.0041 +0.0104γ, γ-Z, mass −0.0003 −0.0008Total +0.0025 +0.0062

Table 8: Corrections to be applied to the quark asymmetries as A0FB = Ameas

FB + δAFB.

The measurements used are:

• Measurements of AbbFB and Acc

FB using leptons from ALEPH [37], DELPHI [38], L3 [39] andOPAL [40]. These analyses measure either Abb

FB only from a high pt lepton sample or they obtainAbb

FB and AccFB from a fit to the lepton spectra. In the case of OPAL the lepton information is

combined with hadronic variables in a neural net. DELPHI uses in addition lifetime informationand jet-charge in the hemisphere opposite to the lepton to separate the different lepton sources.Some asymmetry analyses also measure χ.

• Measurements of AbbFB based on lifetime tagged events with a hemisphere charge measurement

from ALEPH [41], DELPHI [42], L3 [43] and OPAL [44]. These measurements contribute roughlythe same weight to the combined result as the lepton fits.

• Analyses with D mesons to measure AccFB from ALEPH [45] or Acc

FB and AbbFB from DELPHI [46]

and OPAL [47].

• Measurements of Ab and Ac from SLD. These results include measurements using lepton [48,49],D meson [50] and vertex mass plus hemisphere charge [51] tags, which have similar sourcesof systematic errors as the LEP asymmetry measurements. SLD also uses vertex mass forbottom or charm tag in conjunction with a kaon tag or a vertex charge tag for both Ab and Ac

measurements [52–54].

5.1.4 Other Measurements

The measurements of the charmed hadron fractions P(c → D∗+) × BR(D∗+ → π+D0), f(D+), f(Ds)and f(cbaryon) are included in the Rc measurements and are described there.

ALEPH [55], DELPHI [56], L3 [26, 57] and OPAL [58] measure BR(b → `), BR(b → c → ¯) andχ or a subset of them from a sample of leptons opposite to a b-tagged hemisphere and from a doublelepton sample. DELPHI [30] and OPAL [59] measure BR(c→ `) from a sample opposite to a highenergy D∗±.

5.2 Results

5.2.1 Results of the 12-Parameter Fit to the LEP Data

Using the full averaging procedure gives the following combined results for the electroweak parameters:

R0b = 0.21648 ± 0.00075 (17)

16

R0c = 0.1674 ± 0.0047

A0, bFB = 0.0989 ± 0.0020

A0, cFB = 0.0688 ± 0.0035 ,

where all corrections to the asymmetries and partial widths are applied. The χ2/d.o.f. is 49/(89−12).The corresponding correlation matrix is given in Table 9.

R0b R0

c A0, bFB A0, c

FB

R0b 1.00 −0.15 −0.03 0.01

R0c −0.15 1.00 0.07 −0.01

A0, bFB −0.03 0.07 1.00 0.11

A0, cFB 0.01 −0.01 0.11 1.00

Table 9: The correlation matrix for the four electroweak parameters from the 12-parameter fit.

5.2.2 Results of the 14-Parameter Fit to LEP and SLD Data

Including the SLD results for Rb, Rc, Ab and Ac into the fit the following results are obtained:

R0b = 0.21653 ± 0.00069 (18)

R0c = 0.1709 ± 0.0034

A0, bFB = 0.0990 ± 0.0020

A0, cFB = 0.0689 ± 0.0035Ab = 0.922 ± 0.023Ac = 0.631 ± 0.026 ,

with a χ2/d.o.f. of 54/(98 − 14). The corresponding correlation matrix is given in Table 10 and thelargest errors for the electroweak parameters are listed in Table 11.

In deriving these results the parameters Ab and Ac are treated as independent of the forward-backward asymmetries A0, b

FB and A0, cFB. In Figure 2 the results for R0

b and R0c are shown compared with

the Standard Model expectation.

R0b R0

c A0, bFB A0, c

FB Ab Ac

R0b 1.00 −0.13 −0.02 0.01 −0.04 0.02

R0c −0.13 1.00 0.05 −0.01 0.02 −0.02

A0, bFB −0.02 0.05 1.00 0.10 0.02 0.00

A0, cFB 0.01 −0.01 0.10 1.00 0.00 0.01

Ab −0.04 0.02 0.02 0.00 1.00 0.14Ac 0.02 −0.02 0.00 0.01 0.14 1.00

Table 10: The correlation matrix for the six electroweak parameters from the 14-parameter fit.

The 14 parameter fit yields the b → ` branching ratio:

BR(b → `) = 0.1057 ± 0.0019. (19)

17

R0b R0

c A0, bFB A0, c

FB Ab Ac

(10−3) (10−3) (10−3) (10−3) (10−2) (10−2)statistics 0.43 2.6 1.7 3.0 1.6 2.0

internal systematics 0.31 1.8 0.7 1.4 1.6 1.6QCD effects 0.19 0.1 0.2 0.1 0.6 0.3

BR(D → neut.) 0.14 0.1 0 0 0 0D decay multiplicity 0.12 0.2 0 0 0 0BR(D+ → K−π+π+) 0.10 0.3 0.1 0 0 0

BR(Ds → φπ+) 0.02 0.7 0.1 0 0 0BR(Λc →p K−π+) 0.06 0.6 0 0.1 0 0

D lifetimes 0.06 0.1 0 0.1 0 0gluon splitting 0.26 0.6 0 0.2 0.1 0.1c fragmentation 0.10 0.3 0.1 0.2 0.1 0.1

light quarks 0.07 0.3 0.5 0.1 0 0total 0.69 3.4 2.0 3.5 2.3 2.6

Table 11: The dominant error sources for the electroweak parameters from the 14-parameter fit.

The largest error sources on this quantity are the dependences on the semileptonic decay models b → `,c → ` with

∆BR(b → `)|(b→`)model = 0.0008,∆BR(b → `)|(c→`)model = 0.0005.

Extensive studies are made to understand the size of these errors. If all the asymmetry measurementsare excluded from the fit a consistent result is obtained with modelling errors of 0.0010 and 0.0006.The reduction of the modelling uncertainty is due to the inclusion of asymmetry measurements usingdifferent methods. Those using leptons depend on the semileptonic decay models while those usinga lifetime tag and jet charge or D mesons do not. The mutual consistency of the asymmetry mea-surements effectively constrains the semileptonic decay models, and reduces the uncertainty in thesemileptonic branching ratio.

The result of the full fit to the LEP+SLC results including the off-peak asymmetries and thenon-electroweak parameters can be found in Appendix A. Results for the non-electroweak parametersare independent of the treatment of the off-peak asymmetries and the SLD data.

18

0.16

0.17

0.18

0.19

0.214 0.216 0.218 0.22

Rb0

Rc0

Preliminary

68% CL

95% CL

SM

Figure 2: Contours in the R0b-R0

c plane derived from the LEP+SLD data, corresponding to 68%and 95% confidence levels assuming Gaussian systematic errors. The Standard Model prediction formt = 174.3 ± 5.1 GeV is also shown. The arrow points in the direction of increasing values of mt.

19

6 The Hadronic Charge Asymmetry 〈QFB〉

Updates with respect to last summer:L3 have published their result.

The LEP experiments ALEPH [60–62], DELPHI [63, 64], L3 [43] and OPAL [65, 66] have providedmeasurements of the hadronic charge asymmetry based on the mean difference in jet charges measuredin the forward and backward event hemispheres, 〈QFB〉. DELPHI have also provided a related mea-surement of the total charge asymmetry by making a charge assignment on an event-by-event basisand performing a likelihood fit [63]. The experimental values quoted for the average forward-backwardcharge difference, 〈QFB〉, cannot be directly compared as some of them include detector dependenteffects such as acceptances and efficiencies. Therefore the effective electroweak mixing angle, sin2θlept

eff ,as defined in Section 12.4, is used as a means of combining the experimental results summarised inTable 12.

Experiment sin2θlepteff

ALEPH (90-94), final 0.2322 ± 0.0008 ± 0.0011DELPHI (91-94), prel. 0.2311 ± 0.0010 ± 0.0014L3 (91-95), final 0.2327 ± 0.0012 ± 0.0013OPAL (91-94), prel. 0.2326 ± 0.0012 ± 0.0013

LEP Average 0.2321 ± 0.0010

Table 12: Summary of the determination of sin2θlepteff from inclusive hadronic charge asymmetries

at LEP. For each experiment, the first error is statistical and the second systematic. The latter isdominated by fragmentation and decay modelling uncertainties.

The dominant source of systematic error arises from the modelling of the charge flow in thefragmentation process for each flavour. All experiments measure the required charge properties forZ → bb events from the data. ALEPH also determines the charm charge properties from the data.The fragmentation model implemented in the JETSET Monte Carlo program [67] is used by allexperiments as reference; the one of the HERWIG Monte Carlo program [68] is used for comparison.The JETSET fragmentation parameters are varied to estimate the systematic errors. The centralvalues chosen by the experiments for these parameters are, however, not the same. The smaller of thetwo fragmentation errors in any pair of results is treated as common to both. The present averageof sin2θlept

eff from 〈QFB〉 and its associated error are not very sensitive to the treatment of commonuncertainties. The ambiguities due to QCD corrections may cause changes in the derived value ofsin2θlept

eff . These are, however, well below the fragmentation uncertainties and experimental errors.The effect of fully correlating the estimated systematic uncertainties from this source between theexperiments has a negligible effect upon the average and its error.

There is also some correlation between these results and those for AbbFB using jet charges. The

dominant source of correlation is again through uncertainties in the fragmentation and decay modelsused. The typical correlation between the derived values of sin2θlept

eff from the 〈QFB〉 and the AbbFB jet

charge measurements is estimated to be about 20% to 25%. This leads to only a small change in therelative weights for the Abb

FB and 〈QFB〉 results when averaging their sin2θlepteff values (Section 12.4).

Furthermore, the jet charge method contributes at most half of the weight of the AbbFB measurement.

Thus, the correlation between 〈QFB〉 and AbbFB from jet charge will have little impact on the overall

Standard Model fit, and is neglected at present.

20

7 Averages for ff Production at LEP-II

Updates with respect to last summer:Results are updated with data taken in 1999 and 2000.

Since the start of the LEP-II program LEP has delivered collisions at energies from ∼ 130 GeV to∼ 209 GeV. The four LEP experiments have made measurements on the e+e− → ff process over thisrange of energies, and preliminary combinations of these data are discussed in this note.

For the combination presented here, only data taken up to the end of June 2000 are considered.The nominal and actual centre-of-mass energies to which the LEP data are averaged for each year aregiven in Table 13.

A number of measurements on the process e+e− → ff exist and are combined.

• preliminary averages of cross section and forward-backward asymmetry measurements• a preliminary average of the differential cross section measurements, dσ

d cos θ , for the channelse+e− → µ+µ− and e+e− → τ+τ−

• heavy flavour results Rb, Rc, AbbFB and Acc

FB

Complete results of the combinations are available on the web page [69] and are discussed in [70].

The combined results are interpreted in terms of contact interactions, the exchange of Z′ bosons,and contours of the S-Matrix parameters jtotb,c and jfbb,c, that describe γ-Z interference, are derived. Theinterpretations are discussed fully in [70].

7.1 Cross Sections and Asymmetry Measurements

Cross section results are combined for the e+e− → qq, e+e− → µ+µ− and e+e− → τ+τ− channels,forward-backward asymmetry measurements are combined for the µ+µ− and τ+τ− final states. Theaverages are made for the samples of events with high

√s′.

As before [71], the averaged results are given for two signal definitions:

• Definition 1:√s′ is taken to be the mass of the s-channel propagator, with the ff signal being

defined by the cut√s′/s > 0.85. ISR-FSR photon interference is subtracted to render the

propagator mass unambiguous.

• Definition 2: For dilepton events,√s′ is taken to be the bare invariant mass of the outgoing

difermion pair. For hadronic events, it is taken to be the mass of the s-channel propagator.In both cases, ISR-FSR photon interference is included and the signal is defined by the cut√s′/s > 0.85. When calculating the contribution to the hadronic cross section due to ISR-FSR

interference, since the propagator mass is ill-defined, it is replaced by the bare qq mass.

Events containing additional fermion pairs from radiative processes are considered to be signal, pro-viding that the primary pair passes the cut on

√s′/s and that the secondary pair has a mass below

70 GeV/c2.

21

Year Nominal Energy Actual Energy LuminosityGeV GeV pb−1

1995 130 130.2 ∼ 3136 136.2 ∼ 3133∗ 133.2 ∼ 6

1996 161 161.3 ∼ 10172 172.1 ∼ 10167∗ 166.6 ∼ 20

1997 130 130.2 ∼ 2136 136.2 ∼ 2183 182.7 ∼ 50

1998 189 188.6 ∼ 1701999 192 191.6 ∼ 30

196 195.5 ∼ 80200 199.5 ∼ 80202 201.6 ∼ 40

2000 205 204.9 ∼ 60207 206.7 ∼ 30206∗ 205.5 ∼ 90

Table 13: The nominal and actual centre-of-mass energies for data collected during LEP-II operationin each year. The approximate average luminosity analysed per experiment at each energy is alsoshown. Values marked with a ∗ are average energies for 1995, 1996 and 2000 used for heavy flavourresults. The data taken at nominal energies of 130 and 136 in 1995 and 1997 are combined by mostexperiments.

The data are split into 3 sets: data taken at energies from 130–189 GeV, data taken during 1999,and data taken in 2000. Averages are performed separately for each of these data sets. Within eachsubset correlations between experiments and energies and channels are considered.

Tables 14 and 15 show the preliminary combined results for the 1995–1999 data correspondingto the signal definition 1 and the difference in the results if definition 2 is used. The results for theaverages of the 130–189 GeV data are identical to those given in [72]. Results for the more preliminarydata taken during 2000 are not given in numerical form but are shown in Figure 3 which show theLEP averaged cross sections and asymmetries (based on definition 1), respectively, as a function ofthe centre-of-mass energy, together with the SM predictions.

The χ2 per degree of freedom for the average of the 1999 data is 52.5/60. The correlations arerather small, with the largest components at any given pair of energies being between the hadroniccross sections.

There is good agreement between the SM expectations and the measurements of the individualexperiments and the combined averages. The cross sections for hadronic final states at most of theenergy points are somewhat above the SM expectations. Taking into account the correlations betweenthe data points and also assigning a theory error of ±0.2% [73] to the SM predictions, the difference ofthe cross section from the SM expectations averaged over all energies is approximately a 2.5 standarddeviation excess. It is concluded that there is no significant evidence in the results of the combinationsfor physics beyond the SM in the process e+e− → ff.

22

√s (GeV) Quantity Value SM ∆130 σ(qq) [pb] 81.938±2.220 82.803 -0.251

σ(µ+µ−) [pb] 8.592±0.682 8.439 -0.331σ(τ+τ−) [pb] 9.082±0.931 8.435 -0.108Afb(µ+µ−) 0.692±0.060 0.705 0.012Afb(τ+τ−) 0.663±0.076 0.704 0.012

136 σ(qq) [pb] 66.570±1.967 66.596 -0.224σ(µ+µ−) [pb] 8.231±0.678 7.281 -0.280σ(τ+τ−) [pb] 7.123±0.821 7.279 -0.091Afb(µ+µ−) 0.704±0.060 0.684 0.013Afb(τ+τ−) 0.752±0.088 0.683 0.014

Table 14: Preliminary combined LEP results for e+e− → ff for centre-of-mass energies below the W–pair production threshold. All the results correspond to the signal definition 1. The Standard Modelpredictions are from ZFITTER v6.10 [74]. The difference, ∆, in the averages for the measurementsfor definition 2 relative to definition 1 are also indicated. The quoted uncertainties do not include thetheoretical uncertainties on the corrections discussed in [70]

7.2 Differential cross sections

The LEP experiments have measured the differential cross section, dσd cos θ , for the e+e− → µ+µ− and

e+e− → τ+τ− channels. This section discusses a procedure to combine these measurements andpresents preliminary results.

Using a Monte Carlo simulation it is found that a χ2 fit to the measured differential cross sections,using the expected error on the differential cross sections, computed from the expected cross sectionsand the expected numbers of events in each experiment, provided a very good approximation to theexact likelihood method based on Poisson statistics. Further details are given in [70].

Data are binned in 10 bins of cos θ. The scattering angle, θ, is the angle of the negative leptonwith respect to the incoming electron direction in the lab coordinate system. The outer acceptancesof the most forward and most backward bins for which the four experiments have presented their dataare different. This is accounted for as a correction to a common signal definition. The signal definitionused corresponds to definition 1 of Section 7.1.

Correlated small systematic errors between different experiments, channels and energies, arisingfrom uncertainties on the overall normalisation are considered.

The data are subdivided into two energy ranges, 183 and 189 GeV and 192–202 GeV, and averagesare made for each energy point within each of these subsets. The results of the averages are shown inFigures 4 and 5.

The correlations between bins in the average are less that 2% of the total error on the averagesin each bin. The overall agreement between the averaged data and the predictions is good, witha χ2 of 114 for 120 degrees of freedom. At 202 GeV the cross section in the most backward bin,−1.00 < cos θ < −0.8, for both muon and tau final states is above the predictions. For the muonsthe excess in data corresponds to 3.3 standard deviations. For the taus the excess is 2.3 standarddeviations, however, for this measurement the individual experiments are somewhat inconsistent,having a chi-squared with respect to the average of 10.5 for 2 degrees of freedom.

23

√s (GeV) Quantity Value SM ∆161 σ(qq) [pb] 36.909±1.071 35.247 -0.143

σ(µ+µ−) [pb] 4.586±0.364 4.613 -0.178σ(τ+τ−) [pb] 5.692±0.545 4.613 -0.061Afb(µ+µ−) 0.535±0.067 0.609 0.017Afb(τ+τ−) 0.646±0.077 0.609 0.016




192 σ(qq) [pb] 22.292±0.514 21.237 –0.098σ(µ+µ−) [pb] 2.941±0.175 3.097 –0.127σ(τ+τ−) [pb] 2.863±0.216 3.097 –0.047AFB(µ+µ−) 0.540±0.052 0.566 0.019AFB(τ+τ−) 0.610±0.071 0.566 0.019




Table 15: Preliminary combined LEP results for e+e− → ff for centre-of-mass energies above the W–pair production threshold. All the results correspond to the signal definition 1. The Standard Modelpredictions are from ZFITTER v6.10 [74]. The difference, ∆, in the averages for the measurementsfor definition 2 relative to definition 1 are also indicated. The quoted uncertainties do not include thetheoretical uncertainties on the corrections discussed in [70]

24

Cro

ss s

ectio

n (p

b)

√s

´/s

> 0.85

e+e−→hadrons(γ)e+e−→µ+µ−(γ)e+e−→τ+τ−(γ)

LEPpreliminary

√s

(GeV)

σ mea

s/σ S

M

1

10

10 2

0.8

0.9

1

1.1

1.2

120 140 160 180 200 220F

orw

ard-

Bac

kwar

d A

sym

met

ry √s

´/s

> 0.85

e+e−→µ+µ−(γ)e+e−→τ+τ−(γ)

LEPpreliminary

√s

(GeV)

AF

B

mea

s -AF

B

SM

0

0.2

0.4

0.6

0.8

1

-0.2

0

0.2

120 140 160 180 200 220

Figure 3: Preliminary combined LEP results on the cross sections for qq, µ+µ− and τ+τ− final states,and forward-backward asymmetries for µ+µ− and τ+τ− final states as a function of centre-of-massenergy. The values at 130–189 GeV are taken from [72]. The prediction of the Standard Model,computed with ZFITTER [74], are shown as curves. The lower plots shows the ratio of the datadivided by the predictions for the cross sections, and the difference between the measurements andthe predictions for the asymmetries. For better visibility, the µ+µ− and τ+τ− data points are slightlyshifted apart from their nominal centre-of-mass value.

7.3 Heavy Flavour Measurements

A combination of measurements of the ratios3, Rb and Rc and the forward-backward asymmetries,Abb

FB and AccFB, from the LEP collaborations at centre-of-mass energies in the range of 130 to 209 GeV

is performed.

A common signal definition is defined for all the measurements, requiring:

• an effective centre-of-mass energy√s′ > 0.85

√s

• the inclusion of ISR and FSR photon interference contribution and• extrapolation to full angular acceptance.

Systematic errors are divided into three categories: uncorrelated errors, errors correlated betweenthe measurements of each experiment, and errors common to all experiments.

3Unlike at LEP-I, Rq is defined asσqqσhad

.

25

Preliminary LEP Averaged d σ/dcosθ (µµ )

183 GeV

cosθµ

dσ

/dco

sθ (

pb)

189 GeV

cosθµ

dσ

/dco

sθ (

pb)

192 GeV

cosθµ

dσ

/dco

sθ (

pb)

196 GeV

cosθµ

dσ

/dco

sθ (

pb)

200 GeV

cosθµ

dσ

/dco

sθ (

pb)

202 GeV

cosθµ

dσ

/dco

sθ (

pb)

0

2

4

-1 -0.5 0 0.5 10

1

2

3

4

-1 -0.5 0 0.5 1

0

1

2

3

4

-1 -0.5 0 0.5 10

1

2

3

4

-1 -0.5 0 0.5 1

0

1

2

3

4

-1 -0.5 0 0.5 10

1

2

3

-1 -0.5 0 0.5 1

Figure 4: LEP averaged differential cross sections for e+e− → µ+µ− at energies of 183-202 GeV. TheStandard Model Predictions shown as solid histograms are computed with ZFITTER [74]

The results of the combination are presented in Table 16 and Figure 6. Because of the largecorrelation (-0.36) with Rc at 183 GeV and 189 GeV, the errors on the corresponding measurementsof Rb receive an additional contribution which is absent at the other energy points. For other energieswhere there is no measurement of Rc, the Standard Model value of Rc is used in extracting Rb. Theerror that this introduces on Rb is assumed to be negligible. The results are consistent with theStandard Model predictions of ZFITTER.

7.4 Interpretation

The combined cross sections, asymmetries and results on heavy flavour production are interpreted ina variety of models. The cross section and asymmetry results are used to place limits on the mass ofa possible additional heavy neutral boson, Z′. Limits on contact interactions between leptons and oncontact interaction between electrons and b and c quarks are obtained. Heavy flavour results are alsoused within the S-Matrix formalism to give information on the γ-Z interference for heavy quarks.

26

Preliminary LEP Averaged d σ/dcosθ (ττ)

183 GeV

cosθτ

dσ

/dco

sθ (

pb)

189 GeV

cosθτ

dσ

/dco

sθ (

pb)

192 GeV

cosθτ

dσ

/dco

sθ (

pb)

196 GeV

cosθτ

dσ

/dco

sθ (

pb)

200 GeV

cosθτ

dσ

/dco

sθ (

pb)

202 GeV

cosθτ

dσ

/dco

sθ (

pb)

0

2

4

6

-1 -0.5 0 0.5 10

1

2

3

4

-1 -0.5 0 0.5 1

0

1

2

3

4

-1 -0.5 0 0.5 10

2

4

-1 -0.5 0 0.5 1

0

2

4

-1 -0.5 0 0.5 10

1

2

3

-1 -0.5 0 0.5 1

Figure 5: LEP averaged differential cross sections for e+e− → τ+τ− at energies of 183-202 GeV. TheStandard Model Predictions shown as solid histograms are computed with ZFITTER [74]

7.5 Models with Z′ Bosons

The combined hadronic and leptonic cross sections and the leptonic forward-backward asymmetriesare used to fit the data to models including an additional heavy neutral boson, Z′ within a variety ofmodels [77].

Fits are made to the mass of a Z′, MZ′ , for 4 different models referred to as χ, ψ, η and L-R andfor the Sequential Standard Model [78], which proposes the existence of a Z′ with exactly the samecoupling to fermions as the standard Z. LEP-II data alone does not significantly constrain the mixingangle between the Z and Z′ fields, ΘZZ′ . However results from a single experiment where LEP-I datais used in the fit show that the mixing is consistent with zero (see for example [79]). So for these fitsΘZZ′ is fixed to zero.

No evidence is found for the existence of a Z′ boson in any of the models. 95% confidence levellower limits on MZ′ are obtained, by integrating the likelihood function4. The lower limits on the Z′

4To be able to obtain confidence limits from the likelihood function it is necessary to convert the likelihood to a

27

√s (GeV) Rb Rc Abb

FB AccFB

133 0.1809 ±0.0133 - 0.357 ±0.251 0.580 ±0.314(0.1853) - (0.487) (0.681)

167 0.1479 ±0.0127 - 0.618 ±0.254 0.921 ±0.344(0.1708) - (0.561) (0.671)

183 0.1616 ±0.0101 0.270 ±0.043 0.527 ±0.155 0.662 ±0.209(0.1671) (0.250) (0.578) (0.656)

189 0.1559 ±0.0066 0.241 ±0.024 0.500 ±0.096 0.462 ±0.197(0.1660) (0.252) (0.583) (0.649)

192 0.1688 ±0.0187 - 0.371 ±0.302 -(0.1655) - (0.585) -

196 0.1577 ±0.0109 - 0.721 ±0.194 -(0.1648) - (0.587) -

200 0.1621 ±0.0111 - 0.741 ±0.206 -(0.1642) - (0.590) -

202 0.1873 ±0.0177 - 0.591 ±0.284 -(0.1638) - (0.591) -

206 0.1696 ±0.0182 - 0.881 ±0.221 -(0.1633) - (0.593) -

Table 16: Results of the global fit, compared to the Standard Model predictions for the signal definition,computed with ZFITTER [75] in parentheses. Quoted errors represent the statistical and systematicerrors added in quadrature.

mass are shown in Table 17.

Model χ ψ η L-R SSM

MlimitZ′ (GeV/c2) 630 510 400 950 2260

Table 17: 95% confidence level lower limits on the Z′ mass and χ, ψ, η, L-R and SSM models.

7.6 Contact Interactions between Leptons

Following reference [80], contact interactions are parameterised by an effective Lagrangian, Leff , whichis added to the Standard Model Lagrangian and has the form:

Leff =g2

(1 + δ)Λ2

∑i,j=L,R

ηijeiγµeif jγµfj,

where g2/4π is taken to be 1 by convention, δ = 1(0) for f = e (f 6= e), ηij = ±1 or 0, Λ is the scaleof the contact interactions, ei and fj are left or right-handed spinors. By assuming different helicitycoupling between the initial state and final state currents, a set of different models can be definedfrom this Lagrangian [81], with either constructive (+) or destructive (−) interference between theStandard Model process and the contact interactions. The models and corresponding choices of ηij

probability density function; this is done in a Bayesian approach by multiplying by a prior probability function. Simplyintegrating the likelihood is equivalent to multiplying by a uniform prior probability function.

28

0.14

0.16

0.18

0.2

0.22

0.24

0.26

80 100 120 140 160 180 200√s (GeV)

Rb

Rb

LEPpreliminary

√s’/√s > 0.1 , 0.85

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0.32

80 100 120 140 160 180 200√s (GeV)

Rc

Rc

LEPpreliminary

√s’/√s > 0.1 , 0.85

-0.2

0

0.2

0.4

0.6

0.8

1

80 100 120 140 160 180 200√s (GeV)

b-A

sym

met

ry

Ab FB

LEPpreliminary

√s’/√s > 0.1 , 0.85

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

80 100 120 140 160 180 200√s (GeV)

c-A

sym

met

ry

Ac FB

LEPpreliminary

√s’/√s > 0.1 , 0.85

Figure 6: Preliminary combined LEP measurements of Rb, Rc and AbbFB and Acc

FB. Solid lines representthe Standard Model prediction for the signal definition and dotted lines the inclusive prediction.The theoretical predictions are computed with ZFITTER [75]. The LEP-I measurements are takenfrom [1,76].

are given in Table 18. The models LL, RR, VV, AA, LR, RL, V0, A0 are considered here since thesemodels lead to large deviations in the e+e− → µ+µ− and e+e− → τ+τ− channels. The total hadroniccross section on its own is not particularly sensitive to contact interactions involving quarks. For thepurpose of fitting contact interaction models to the data, a new parameter ε = 1/Λ2 is defined; ε = 0in the limit that there are no contact interactions. This parameter is allowed to take both positiveand negative values in the fits.

The averaged measurements of the cross sections and forward-backward asymmetries for e+e− →µ+µ− and e+e− → τ+τ− from all energies from 130 to 207 GeV are used. Theoretical uncertaintieson the SM predictions of ±0.5% [82] on the cross sections and ±0.005 on the forward-backwardasymmetries, fully correlated between all energies, are assumed.

The values of ε extracted for each model are all compatible with the Standard Model expectationε = 0, at the two standard deviation level. These errors on ε are typically a factor of two smallerthan those obtained from a single LEP experiment with the same data set. The fitted values of εare converted into 95% confidence level lower limits on Λ. The limits are obtained by integratingthe likelihood function over the physically allowed values, ε ≥ 0 for each Λ+ limit and ε ≤ 0 for Λ−

limits. The fitted values of ε and the extracted limits are shown in Table 19. Figure 7 shows the limitsobtained on the scale Λ for the different models assuming universality between contact interactions

29

Model ηLL ηRR ηLR ηRL

LL± ±1 0 0 0RR± 0 ±1 0 0VV± ±1 ±1 ±1 ±1AA± ±1 ±1 ∓1 ∓1LR± 0 0 ±1 0RL± 0 0 0 ±1V0± ±1 ±1 0 0A0± 0 0 ±1 ±1

Table 18: Choices of ηij for different contact interaction models.

LEP Combined Preliminary

Λ- (TeV) Λ+ (TeV)

LL 10.0 15.2

RR 9.1 15.6

VV 15.3 23.9

AA 15.6 18.8

RL 8.0 11.6

LR 8.0 11.6

V0 13.8 22.7

A0 11.0 16.2

Λ- Λ+

ll 20. 0 20.

Figure 7: The limits on Λ for e+e− → `+`− assuming universality in the contact interactions betweene+e− → µ+µ− and e+e− → τ+τ−.

for e+e− → µ+µ− and e+e− → τ+τ−.

7.7 Contact Interactions from Heavy Flavour Averages

Limits on contact interactions between electrons and b and c quarks are obtained. These results areof particular interest since they are inaccessible to pp or ep colliders. The formalism for describingcontact interactions including heavy flavours is identical to that described above for leptons.

All heavy flavour LEP-II combined results from 133 to 205 GeV listed in Table 16 are used asinputs. For the purpose of fitting contact interaction models to the data, Rb and Rc are converted tocross sections σbb and σcc using the averaged qq cross section of section 7.1 corresponding to signal

30

e+e− → µ+µ−

Model ε (TeV−2) Λ−(TeV) Λ+(TeV)

LL −0.0066+0.0039−0.0042 8.2 14.3

RR −0.0069+0.0045−0.0054 8.0 13.4

VV −0.0023+0.0017−0.0018 13.7 21.6

AA −0.0033+0.0032−0.0012 13.1 19.2

RL −0.0052+0.0067−0.0074 7.2 10.1

LR −0.0052+0.0067−0.0074 7.2 10.1

V0 −0.0036+0.0024−0.0022 11.9 20.6

A0 −0.0027+0.0035−0.0033 10.8 14.4

e+e− → τ+τ−


LL −0.0005+0.0057−0.0055 9.5 9.8

RR −0.0005+0.0060−0.0063 8.7 9.5

VV −0.0008+0.0023−0.0036 14.4 16.1

AA −0.0008+0.0033−0.0016 13.4 12.2

RL −0.0052+0.0093−0.0102 6.5 8.8

LR −0.0052+0.0093−0.0102 6.5 8.8

V0 −0.0003+0.0029−0.0029 12.9 13.7

A0 −0.0026+0.0049−0.0050 9.5 12.4

e+e− → l+l−


LL −0.0046+0.0038−0.0036 10.0 15.2

RR −0.0046+0.0038−0.0044 9.1 15.6

VV −0.0019+0.0024−0.0012 15.3 23.9

AA −0.0013+0.0018−0.0015 15.6 18.8

RL −0.0052+0.0054−0.0060 8.0 11.6

LR −0.0052+0.0054−0.0060 8.0 11.6

V0 −0.0023+0.0018−0.0020 13.8 22.7

A0 −0.0027+0.0028−0.0028 11.0 16.2

Table 19: Fitted values of ε and 95% confidence limits on the scale, Λ, for constructive (+) anddestructive interference (−) with the Standard Model, for the contact interaction models discussed inthe text. Results are given for e+e− → µ+µ−, e+e− → τ+τ− and e+e− → `+`−, assuming universalityin the contact interactions between e+e− → µ+µ− and e+e− → τ+τ−.

definition 2. In the calculation of errors, the correlations between Rb, Rc and σqq are assumed to benegligible.

The fitted values of ε = 1Λ2 and their 68% confidence level uncertainties together with the 95%

confidence level lower limits on Λ are shown in Table 20. Figure 8 shows the limits obtained on thescale, Λ, of models with different helicity combinations involved in the interactions.

31

e+e− → bb

Model ε (TeV−2) Λ− (TeV) Λ+ (TeV)LL −0.0025+0.0049

−0.0052 9.1 11.1RR −0.1890+0.1290

−0.0151 2.2 7.2VV −0.0020+0.0041

−0.0043 10.0 12.4AA −0.0018+0.0032

−0.0034 11.2 14.0RL 0.0190+0.1299

−0.0201 7.3 2.4LR −0.0428+0.0408

−0.0367 3.2 5.7V0 −0.0018+0.0035

−0.0037 10.8 12.9A0 0.0266+0.0234

−0.0255 6.4 4.1

e+e− → cc

Model ε (TeV−2) Λ− (TeV) Λ+ (TeV)LL 0.0127+0.5957

−0.0264 5.2 1.6RR 0.0466+0.3781

−0.0576 4.5 1.5VV −0.0008+0.0109

−0.0103 7.3 6.6AA 0.0046+0.0168

−0.0151 6.4 5.1RL 0.0127+0.0845

−0.0845 2.8 2.6LR 0.0874+0.1049

−0.1127 3.5 2.1V0 0.0036+0.0181

−0.0135 6.7 1.4A0 0.0499+0.0691

−0.0691 3.9 2.6

Table 20: Fitted values of ε and 95% confidence limits on the scale, Λ, for constructive (+) anddestructive interference (−) with the Standard Model, for the contact interaction models discussed inthe text. From combined bb and cc results with centre-of-mass energies from 133 to 205 GeV.


Λ− (TeV) Λ+ (TeV)

LL 9.1 11.1

RR 2.2 7.2

VV 10.0 12.4

AA 11.2 14.0

RL 7.3 2.4

LR 3.2 5.7

V0 10.8 12.9

A0 6.4 4.1

Λ− Λ+

bb 20. 0 20.


Λ− (TeV) Λ+ (TeV)

LL 5.2 1.3

RR 4.5 1.5

VV 7.3 6.6

AA 6.4 5.1

RL 2.9 2.6

LR 3.5 2.1

V0 6.7 1.4

A0 3.9 2.6

Λ− Λ+

cc 20. 0 20.

Figure 8: 95% CL limits on the scale of contact interactions in e+e− → bb and e+e− → cc usingHeavy Flavour LEP combined results from 133 to 205 GeV.

32

7.8 S-Matrix Parameters for Heavy Flavour Production

The S-Matrix formalism [83] parameterises the cross sections and forward-backward asymmetries fortwo-fermion production in terms of the exchange of a massless (γ) and a massive vector boson (Z):

σ0,ftot (s) =

43πα2 +

[gtotf

s+jtotf (s−m2

Z) + rtotf s

(s −m2Z)2 +m2

ZΓ2Z

]

σ0,ffb (s) =

43πα2 +

[gfbf

s+jfbf (s−m2

Z) + rfbf s

(s−m2Z)2 +m2

ZΓ2Z

]

A0,ffb (s) =

34σ0,f

fb (s)

σ0,ftot (s)

The mass, mZ, and width, ΓZ, used in the S-Matrix fits are slightly different from the usual mass mZ,and width ΓZ which are defined using an s-dependent width term in the Breit-Wigner resonance ofthe Z:

mZ ∼ mZ − 34.1MevΓZ ∼ ΓZ − 0.9Mev

The parameters g, r and j parameterise the cross sections and forward-backward asymmetries arisingfrom the exchange of the γ (g) and the Z (r) and the interference of the two (j). Values for theseparameters can be obtained for each fermion species and also for a combination of all qq final states,f = had. Values of the g, r and j parameters and ΓZ are computed in the Standard Model forcomparison with the results of the fits.

S-Matrix fits have already been performed using the hadronic and leptonic cross section andleptonic forward-backward asymmetry data from LEP-I and LEP-II for energies up to 172 GeV [71],providing constraints on rtothad and jtothad, the parameters describing charged lepton production andmZ and ΓZ. The results of these existing fits and the full error matrix are used to constrain theseparameters in the fit performed here.

LEP-I and SLD heavy flavour averages [84], Rb, Rc, AbbFB and Acc

FB are used to constrain theS-Matrix parameters rtotb,c and rfbb,c that describe the heavy quark couplings to the Z. By includingthe combined LEP-II heavy flavour measurements from 133 to 205 GeV listed in Table 16 values ofthe parameters jtotb,c and jfbb,c that describe γ-Z interference in heavy quark production are obtained.Contours for these parameters are given at the end of this section.

The LEP-II average values of Rb and Rc from Section 7.3 for centre-of-mass energies from 130up to 166 GeV are used directly in the fit. These are largely uncorrelated with the measured totalcross sections, therefore, the correlations between the existing S-Matrix fits parameters and thesemeasurements are neglected. The measurements are fitted using the ratio of the predicted b and ccross sections and the total hadronic cross section.

For energies of 183 GeV and above, the flavour tagged measurements are more precise than thosefrom 133–166 GeV. The uncertainties on jtothad from the existing fits introduces a sizeable uncertaintyon the prediction of Rb and Rc. For these energies the measurements of Rb and Rc are first convertedinto cross sections for bb and cc production by multiplying by the the total hadronic cross sections

33

from section 7.1. The full error matrix of the quantities and the correlations with the lower energyRb and Rc values are computed. Correlations between the existing S-Matrix fit results and the totalhadronic cross sections from 183 GeV and above are neglected. For this reason the highest energyhadronic cross sections are not used to improve the fits to the inclusive hadronic S-Matrix parameters.

The forward-backward asymmetries for b and c quark production are fitted directly to predictionsof the asymmetries at all energies.

Predictions of the S-Matrix formalism are made using the SMATASY [85] program. The couplingsto the photon are fixed to the expectation from the Standard Model.

In summary, the following parameters are obtained from the fit:

• The mass and total width of the Z boson, the S-Matrix parameters rtothad and jtothad for the totalhadronic cross section, and the parameters rtotl , jtotl , rfbl , and jfbl for leptons5.

• The S-Matrix parameters rtotb,c and rfbb,c that describe the total bb and cc cross sections andasymmetries due to Z boson exchange.

• The four parameters jtotb,c and jfbb,c that describe the the effect of γ-Z interference on the energydependence of the bottom and charm cross sections and asymmetries.

The following data are used:

• The results of the existing S-Matrix fits [71], derived from LEP-I data and including totalhadronic cross sections up to 172 GeV.

• LEP-I and SLD heavy flavour averages [84].• LEP-II heavy flavour averages Rb andRc for energies up to and including 172 GeV from Table 16.• Values of the flavour tagged b and c cross sections σb and σc derived from measurements of Rb

and Rc and the total hadronic cross section for energies of 183 GeV and above, from section 7.1and Table 16.

• LEP-II heavy flavour averages AbbFB and Acc

FB for all LEP-II energies from Table 16.

The results for the S-Matrix parameters jtotb,c and jfbb,c are shown in Figure 9 for both bottom andcharm quark production. Good agreement is observed with the Standard Model prediction [85] forγ-Z interference in heavy quark production.

5Lepton universality is assumed.

34

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

-0.06-0.04-0.02 0 0.02 0.04 0.06 0.08 0.1jb

tot

j b fb

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.1

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4j c tot

j c fb

Figure 9: Preliminary combined LEP results on the S-Matrix parameters describing γ-Z interferencein bb and cc production. The expectations of the SM are also shown.

35

8 Measurement of W Boson Properties at LEP-II

Updates with respect to last summer:There are new results on the WW cross section and on the W mass. The W decay branching ratiosare updated.

In the year 2000 LEP ran at centre-of-mass energies larger than 200 GeV, with a maximum of 209 GeV(see Table 13). The collected data are divided in two ranges of

√s, below and above 205.5 GeV. The

two data sets have mean centre-of-mass energies of 204.9 and 206.7 GeV and the respective integratedluminosities used for the analyses considered in this note are 60 and 30 pb−1. The same centre-of-massenergy binning is also applied for the measurement of the Z–pair cross section. All data recorded from1996 to 1999 over a range of centre-of-mass energies,

√s = 161− 202 GeV are used to extract the W

boson mass and width.

8.1 W–pair Production Cross Section

All experiments have published final results on the W–pair (CC03) production cross section for centre-of-mass energies up to 189 GeV [86–89]6. ALEPH [91], DELPHI [92] and L3 [93] preliminary resultsat√s = 192–202 GeV are unchanged with respect to combination of results prepared for the year 2000

winter conferences. OPAL has updated its preliminary results at√s = 200 and 202 GeV [94] to include

the full luminosity collected in 1999, whereas the results at√s = 192 and 196 GeV [95] are unchanged

since the winter conferences. All experiments have contributed preliminary results [94, 96–98] basedon the analysis of year 2000 data.

Results from different experiments are combined using the method described in [99] and takinginto account, when relevant, the correlations between the systematic uncertainties, which arise mainlyfrom the use of the same Monte Carlo programs to predict background cross sections and to simulatethe hadronisation processes.

The results from each experiments for the W–pair production cross section, assuming StandardModel values for the W decay branching ratios, are shown in Table 21, together with the LEP combi-nation. In the averaging procedure the QCD component of the systematic errors 7 from each individualmeasurement is taken to be fully correlated between experiments. This common error ranges between0.07 and 0.10 pb. These results supersede the ones presented in [100] for

√s = 189, 200 and 202 GeV.

Figure 10 shows the total W–pair cross section measured as a function of the LEP centre-of-mass energy. The experimental points are compared with new calculations (RacoonWW [101] andYFSWW3 [102]) based on the double pole approximation (DPA) [103] for mW = 80.35 GeV. These twocodes have been extensively compared and agree at a level better than 0.2% at the LEP-II energies.The theoretical uncertainty of the DPA decreases from 0.7% at 170 GeV to a level of 0.4% at centre-of-mass energies larger than 200 GeV8. This theoretical uncertainty is represented by the grey bandin Figure 10. An error of 50 MeV on the W mass translates into a 0.1% error on the cross sectionpredictions at 200 GeV. The DPA is only valid away from the production threshold, therefore below

6In the combination the preliminary results of the L3 collaboration [90] are still used, as the final results only becameavailable shortly before the ICHEP conference.

7The QCD component of the systematic error, includes for example uncertainties on the 4–jet rate, on the fragmen-tation modelling and estimates of possible effects of colour reconnection and Bose-Einstein correlations.

8The theoretical uncertainty ∆σ/σ on the W–pair production cross section calculated in the DPA can be parametrisedas ∆σ/σ = 0.4⊕ 0.072 · t1 · t2, where t1 = (200− 2mW)/(

√s− 2mW) and t2 = (1− ( 2·mW

200)2)/(1− ( 2·mW√

s)2) [104].

36

170 GeV the experimental results are compared with the Gentle [105] prediction, which is adjustedto reproduce the DPA results at higher energies. All results, up to the highest centre-of-mass energies,are in agreement with the theoretical predictions.

√s Cross section (pb) χ2/d.o.f.

(GeV) ALEPH DELPHI L3 OPAL LEP161.33 4.23 ± 0.75∗ 3.67± 0.93∗ 2.89± 0.77∗ 3.62 ± 0.89∗ 3.69 ± 0.45 1.3/3172.12 11.7 ± 1.3∗ 11.58 ± 1.4∗ 12.27 ± 1.4∗ 12.3 ± 1.3∗ 12.0 ± 0.7 0.22/3182.67 15.57 ± 0.68∗ 15.86 ± 0.74∗ 16.53 ± 0.72∗ 15.43 ± 0.66∗ 15.83 ± 0.36 1.49/3188.63 15.71 ± 0.38∗ 15.83 ± 0.43∗ 16.20 ± 0.46 16.30 ± 0.38∗ 16.00 ± 0.21 1.57/3191.6 17.23 ± 0.91 16.90 ± 1.02 16.39 ± 0.93 16.60 ± 0.97 16.78 ± 0.48 0.47/3195.5 17.00 ± 0.57 17.86 ± 0.63 16.67 ± 0.60 18.59 ± 0.74 17.39 ± 0.32 5.26/3199.5 16.98 ± 0.56 17.35 ± 0.60 16.94 ± 0.62 16.32 ± 0.66 16.93 ± 0.31 1.39/3201.6 16.16 ± 0.76 17.67 ± 0.84 16.95 ± 0.88 18.48 ± 0.91 17.20 ± 0.43 4.27/3204.9 16.70 ± 0.64 18.81 ± 0.80 17.70 ± 0.86 15.84 ± 0.71 17.11 ± 0.38 8.79/3206.7 17.01 ± 0.88 16.50 ± 1.05 17.20 ± 1.03 15.96 ± 0.96 16.68 ± 0.49 1.01/3

Table 21: W–pair production cross sections from the four LEP experiments and combined valuesfor the seven of the recorded centre-of-mass energies. All results are preliminary with the exceptionof those indicated by ∗. A common systematic error of (0.07–0.10) pb is taken into account in theaveraging procedure. Final results for the combined W–pair production cross section at lower centre-of-mass energies can be found in [1, 71,106] .

8.2 W Decay Branching Fractions

From the cross sections for the individual W decay channels measured by the four experiments atall energies larger than 161 GeV, the W decay branching ratios (B(W → ff′)) are determined, withand without the assumption of lepton universality. ALEPH [96] and OPAL [94] have updated theirresults to include the data collected in the year 2000 at centre-of-mass energies up to 208 GeV, whereasDELPHI [92] and L3 [93] have not updated their preliminary results prepared for the year 2000 winterconferences. The results from the single experiments are given in Table 22 together with the result ofthe LEP combination. Correlated errors between the different channels of individual experiments aretaken into account in the averaging procedure, together with the QCD component of the systematicerror which is assumed to be fully correlated among experiments.

The results of the fit which does not make use of the lepton universality assumption show anegative correlation of 21.0% (18.7%) between the W → τντ and W → eνe (W → µνµ) branchingratios, while between the electron and muon decay channels there is a positive correlation of 4.4%.These branching ratios are all consistent within the errors and allow to test lepton universality in thedecay of on–shell W bosons at the level of 3.3%. Assuming lepton universality, the measured hadronicbranching ratio is (67.78±0.32)% and the leptonic one is (10.74±0.10)%. These results are consistentwith the Standard Model expectations. Statistical and systematic uncertainties give approximatelyequal contributions to the total errors on the branching ratios. The systematic error is divided equallyin the two components which are correlated and uncorrelated between experiments.

Within the Standard Model, the branching fractions of the W boson depend on the six matrixelements |Vqq′ | of the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix not involving thetop quark. In terms of these |Vqq′ | the leptonic branching ratio of the W boson, B(W → `ν`), is givenby

37

0

5

10

15

20

160 170 180 190 200 210

Ecm [GeV]

σWW

[pb]

LEP Preliminary21/07/2000

Gentle 2.1 (±0.7%)

RacoonWW / YFSWW 1.14

0

5

10

15

20

160 170 180 190 200 2100

5

10

15

20

160 170 180 190 200 210

RacoonWWYFSWW 1.14

16

17

18

Figure 10: Measurements of the W–pair production cross section compared to the predictions ofRacoonWW, and YFSWW3 and Gentle for mW = 80.35 GeV. The shaded area represent theuncertainty on the prediction which is ±0.7% for

√s < 170 GeV and varies between ±0.4 and ±0.7%

for√s > 170 GeV.

1/B(W → `ν`) = 3{1 + [1 + αs(mW2)/π]

∑i=u,c, j=d,s,b

|Vij |2}, (20)

where αs(mW2) is the strong coupling constant. Taking αs(mW

2) = (0.121 ± 0.002), the measuredleptonic branching ratio of the W yields

38

Lepton Leptonnon–universality universality

Experiment W → eνe W → µνµ W → τντ W → hadrons(%) (%) (%) (%)

ALEPH 11.19 ± 0.34 11.05 ± 0.32 10.53 ± 0.42 67.22 ± 0.53DELPHI 10.33 ± 0.45 10.68 ± 0.34 11.28 ± 0.56 67.81 ± 0.61

L3 10.22 ± 0.36 9.87 ± 0.38 11.64 ± 0.51 68.47 ± 0.59OPAL 10.52 ± 0.37 10.56 ± 0.35 11.15 ± 0.49 67.86 ± 0.62LEP 10.62 ± 0.20 10.60 ± 0.18 11.07 ± 0.25 67.78 ± 0.32

χ2/d.o.f. 11.1/9 13.2/9

Table 22: Summary of W branching ratios derived from W–pair production cross sections measure-ments up to 208 GeV centre-of-mass energy (DELPHI and L3 results are based only on data collectedup to

√s = 202 GeV). A common systematic error of (0.04–0.13)% is taken into account in the

averaging procedure.

∑i=u,c, j=d,s,b

|Vij |2 = 2.026 ± 0.030 (Br.) ± 0.001 (αs), (21)

where the first error is due to the uncertainty on the branching ratio measurement and the second tothe uncertainty on αs. Using the experimental knowledge of the sum |Vud|2 + |Vus|2 + |Vub|2 + |Vcd|2 +|Vcb|2 = (1.0477 ± 0.0074) [107], the above result can be interpreted as a measurement of |Vcs| whichis the least well determined of these matrix elements:

|Vcs| = 0.989 ± 0.016. (22)

The error includes the uncertainties on αs and on the other |Vqq′ | matrix elements but is dominatedby the experimental error on the W branching fractions. The uncertainty in the sum of the other CKMmatrix elements, which is dominated by the uncertainty on |Vcd|, contributes a negligible uncertaintyof 0.004 to this determination of |Vcs|.

8.3 W Mass Measurement

The W boson mass results presented here are obtained from data recorded over a range of centre-of-mass energies,

√s = 161 − 202 GeV during the 1996-1999 operation of the LEP collider. These data

correspond to an integrated luminosity of approximately 460 pb−1 per experiment. No results from theyear 2000 LEP running are included. The results on the W mass quoted below correspond to a massdefinition based on a Breit-Wigner denominator with an s-dependent width, |s−mW

2 + isΓW/mW|.

Since 1996 the LEP e+e− collider is operating above the threshold for W+W− pair production.Initially 10 pb−1 of data were recorded close to W+W− pair production threshold. At this energy theW+W− cross section is sensitive to the W boson mass, mW. Table 23 summarises the W mass resultsfrom the four LEP collaborations based on these data [108–111].

Subsequently LEP has operated well above the W+W− threshold, delivering 450 pb−1 per experi-ment (excluding the operation in 2000). Above threshold, the e+e− → W+W− cross section has little

39

Threshold Cross SectionExperiment mW/GeV

ALEPH [108] 80.14 ± 0.35DELPHI [109] 80.40 ± 0.45

L3 [110] 80.80+0.48−0.42

OPAL [111] 80.40+0.46−0.43

Table 23: W mass measurements from the W+W− threshold cross section at√s = 161 GeV. The

errors include statistical and systematic contributions.

sensitivity tomW. For these data the W boson mass is measured using the method of direct reconstruc-tion where the measured four-momenta of the reconstructed jets and leptons are used to reconstructthe invariant mass on an event-by-event basis. Table 24 summarises the W mass results from the fourLEP experiments using the method of direct reconstruction. In addition to the combined numbers,each experiment presents mass measurements from W+W−→qq`ν` and W+W−→qqqq channels sep-arately. The DELPHI, L3 and OPAL collaborations obtain results for the qq`ν` and qqqq channelsusing independent fits to data from each channel, taking into account correlated systematic errors. Theqq`ν` and qqqq results quoted by the ALEPH collaboration are obtained from a simultaneous fit to alldata which, in addition to other correlations, takes into account the correlated systematic uncertaintiesbetween the two channels. The large variation in the systematic uncertainties in the W+W−→qqqqchannel are caused by differing estimates of the possible effects of colour reconnection (CR) and Bose-Einstein correlations (BEC), discussed below. The systematic errors in the W+W−→qq`ν` channelare dominated by uncertainties from hadronisation, with estimates ranging from 20-40 MeV.

DIRECT RECONSTRUCTIONW+W−→qq`ν` W+W−→qqqq Combined

Experiment mW/GeV mW/GeV mW/GeVALEPH [112–114] 80.435 ± 0.063 ± 0.048 80.467 ± 0.064 ± 0.057 80.449 ± 0.045 ± 0.047

DELPHI [115–118] 80.381 ± 0.088 ± 0.048 80.372 ± 0.064 ± 0.063 80.380 ± 0.053 ± 0.047L3 [119–122]] 80.273 ± 0.089 ± 0.046 80.461 ± 0.077 ± 0.069 80.362 ± 0.058 ± 0.052

OPAL [123–127] 80.510 ± 0.067 ± 0.031 80.408 ± 0.066 ± 0.100 80.486 ± 0.053 ± 0.039

Table 24: Preliminary W mass measurements from direct reconstruction (√s = 172 − 202 GeV).

Results are given for the semileptonic, fully hadronic channels and the combined value. The errors arestatistical and systematic respectively. The combined values of mW from each collaboration take intoaccount the correlated systematic uncertainties between the two channels and between the differentyears of data taking. The W+W−→qq`ν` results from the ALEPH and OPAL collaborations includemass information from the W+W− → `−ν``

′+ν`′ channel.

8.3.1 Combination Procedure

A LEP combined W mass measurement is obtained from the results of the four experiments. Tocombine the measurements, a more detailed input than that given in Table 24 is required. Eachexperiment provides a W mass measurement for both W+W−→qq`ν` and W+W−→qqqq channelsfor each of the four years data taking (1996-1999), a total of 30 measurements (the L3 Collaborationprovides results from the 1996 and 1997 data already combined). The subdivision into years enables aproper treatment of the correlated systematic uncertainty from the LEP beam energy and allows for the

40

possibility that the systematic uncertainties depend on the centre-of-mass energy and/or data-takingperiod. For each result a detailed breakdown of the sources of systematic uncertainty is provided andthe correlations are specified. In the combination these inter-year, inter-channel and inter-experimentcorrelations are included. The main sources of correlated systematic errors are: colour reconnection,Bose-Einstein correlations, hadronisation, the LEP beam energy, and uncertainties from initial andfinal state radiation. The full correlation matrix for the LEP beam energy is employed [128].

The four LEP collaborations give different estimates of the systematic errors arising from final stateinteractions, ranging from 30-66 MeV for colour reconnection and from 20-67 MeV for Bose-Einsteincorrelations. These spreads could be due to different experimental sensitivities to these effects or,alternatively, they could be a reflection of the different models used to assess the uncertainties. Thisquestion is addressed using common Monte Carlo samples with and without CR effects. These arepassed through the full detector simulations of each of the four experiments. Studies of these samplesdemonstrate that the four experiments are equally sensitive to colour reconnection effects, i.e. whenlooking at the same CR model similar biases are seen by all experiments. For this reason a commonvalue of the CR systematic uncertainty is used in the combination. For Bose-Einstein Correlations, nosimilar test is made of the respective experimental sensitivities. However, in the absence of evidencethat the experiments have different sensitivities to the effect, a common value of the systematicuncertainty from BEC is assumed. In the combination a common colour reconnection error of 50 MeVand a common Bose-Einstein systematic uncertainty of 25 MeV are used. These uncertainties arerepresentative of the numbers estimated by the four LEP collaborations.

8.3.2 LEP Combined W Boson Mass

The combined W mass from direct reconstruction, taking into account all correlations including thosebetween the W+W−→qq`ν` and W+W−→qqqq channels, years and experiments gives

mW(direct) = 80.428 ± 0.030(stat.)± 0.036(syst.) GeV,

with a χ2/d.o.f. of 27.1/29, corresponding to a χ2 probability of 57%. The weight of the fully hadronicchannel in the combined fit is 0.27. The reduced weight is a consequence of the relatively large size ofthe current estimates of the systematic errors from CR and BEC. Table 25 gives a breakdown of thecontribution to the total error of the systematic errors from different sources and the contribution tototal error from statistics. The largest contribution to the systematic error comes from hadronisationuncertainties, which are treated as correlated between the two channels. In the absence of systematiceffects the current LEP statistical precision on mW would be 25 MeV. The statistical error contributionto the total error from the LEP combination is larger than this (30 MeV) due to the significantlyreduced weight of the fully hadronic channel. Table 25 also shows the error breakdown for W massmeasurements from the two channels separately (described in the following section).

In addition to the above results, the W boson mass is measured at LEP from the data recordedat threshold (10 pb−1/experiment) for W pair production:

mW(threshold) = 80.40 ± 0.20(stat.)± 0.07(syst.)± 0.03(LEP) GeV.

When this is combined with the much more precise results obtained from direct reconstruction oneobtains a W mass measurement of

mW = 80.427 ± 0.046 GeV.

In combining the threshold and direct reconstruction measurements the only correlated systematicuncertainty is that from the LEP beam energy.

41

Source Systematic Error on mW (MeV)qq`ν` qqqq Combined

ISR/FSR 8 10 8Hadronisation 26 23 24Detector Systematics 11 7 10LEP Beam Energy 17 17 17Colour Reconnection − 50 13Bose-Einstein Correlations − 25 7Other 5 5 4Total Systematic 35 64 36Statistical 38 34 30Total 51 73 47

Table 25: Error decomposition for the combined W mass results. Detector systematics include un-certainties in the jet and lepton energy scales and resolution. The ‘Other’ category refers to errors,all of which are uncorrelated between experiments, arising from: Monte Carlo statistics, background,four-fermion treatment, fitting and event selection. The error decomposition in the qq`ν` and qqqqchannels refers to the independent fits to the results from the two channels separately.

8.3.3 Consistency Checks

In addition to fitting for the combined W mass from the W+W−→qq`ν` and W+W−→qqqq channels,the difference between the W boson mass measurements obtained from the fully hadronic and semilep-tonic channels, ∆mW(qqqq− qq`ν`), can be determined. A significant non-zero value for ∆mW couldindicate that FSI effects are biasing the value of mW determined from W+W−→qqqq events. Since∆mW is primarily of interest as a check of the possible effects of final state interactions, the errorsfrom CR and BEC are set to zero in its determination, giving:

∆mW(qqqq− qq`ν`) = +5± 51 MeV.

This result is obtained from a fit to the results of the four experiments taking into account correlationsbetween: the experiments, the two channels and different years. This result is almost unchanged if thesystematic part of the error on mW from fragmentation effects is considered as uncorrelated betweenexperiments and channels, although the uncertainty increases by about 17%. The masses from thetwo channels obtained from this fit are

mW(W+W−→qq`ν`) = 80.427 ± 0.038(stat.)± 0.035(syst.) GeV,mW(W+W−→qqqq) = 80.432 ± 0.034(stat.)± 0.064(syst.) GeV.

These two results are correlated having a correlation coefficient of 0.29. The value of χ2/d.o.f is27.1/28, corresponding to a χ2 probability of 52%. These results and the correlation between themcan be used to combine the two measurements or to form the mass difference.

Experimentally, separatemW measurements are obtained from the W+W−→qq`ν` and W+W−→qqqqchannels. The combination using only the qq`ν` measurements yields

mindepW (W+W−→qq`ν`) = 80.429 ± 0.038(stat.)± 0.035(syst.) GeV.

This result is independent of the measurements in the W+W−→qqqq channel. The systematic error isdominated by fragmentation uncertainties (±26 MeV) and the uncertainty in the LEP centre-of-mass

42

energy (±17 MeV). The independent average over all qqqq measurements only gives

mindepW (W+W−→qqqq) = 80.424 ± 0.034(stat.)± 0.064(syst.) GeV.

where the dominant contributions to the systematic error arise from BE/CR (±56 MeV), fragmentation(±23 MeV) and from the uncertainty in the LEP centre-of-mass energy (±17 MeV).

8.3.4 LEP Combined W Boson Width

The method of direct reconstruction is also used for the direct measurement of the width of the Wboson. The results of the four LEP experiments are shown in Table 26.

Experiment ΓW/GeVALEPH 2.17 ± 0.16 ± 0.12DELPHI 2.09 ± 0.12 ± 0.09L3 2.19 ± 0.14 ± 0.16OPAL 2.04 ± 0.16 ± 0.09Combined 2.12 ± 0.08 ± 0.07

Table 26: Preliminary W width measurements (√s = 172−202 GeV) from the individual experiments.

The first error is statistical and the second systematic. The ALEPH results at 192-202 GeV do notuse the W+W−→qqqq channel and the OPAL results do not include data recorded at centre-of-massenergies greater than 189 GeV. Only OPAL and L3 include data from 172 GeV.

A simultaneous fit to the results of the four LEP collaborations is performed in the same way asfor the mW measurement. Correlated systematic uncertainties are taken into account to give:

ΓW = 2.12 ± 0.08(stat.)± 0.07(syst.) GeV,

with a χ2/d.o.f. of 13.7/17.

Currently no LEP combined value for ΓW is obtained from the W+W−→qq`ν` and W+W−→qqqqchannels separately.

8.3.5 Summary

The results of the four LEP experiments on the mass and width of the W boson are combined takinginto account correlated systematic uncertainties, giving:

mW = 80.427 ± 0.046 GeV,ΓW = 2.12± 0.11 GeV.

43

9 Single–W Production Cross Section

Updates with respect to last summer:This is a new section.

Preliminary single–W results for data collected at√s = 183–202 GeV are available. No preliminary

analysis of the data collected in the year 2000 at√s > 200 GeV was prepared in time for the summer

conferences.

A common definition of the single–W production cross section is adopted, to allow direct compar-isons and the combination of the results obtained by the four experiments. Single–W production isdefined by the complete t-channel subset of Feynman diagrams contributing to the eνff ′ final states,with additional cuts on kinematic variables to exclude the regions of the phase space dominated bymultiperipheral diagrams, where the cross section calculation is affected by large uncertainties. Theadvantage of this definition is that a gauge invariant set of diagrams is taken, which has a small inter-ference with the s-channel one. In addition, no angular cuts due to the specific detector acceptancesare used.

The kinematic cuts used in the cross section definition are (charge conjugation is implied in thedefinition):

• Mqq′ > 45 GeV for the eνqq′ final states,

• E`+ > 20 GeV for the eν`ν final states (with ` = µ, τ),

• |cos(θe−)| > 0.95, |cos(θe+ )| < 0.95, Ee+ > 20 GeV for the eνeν final state.

The results of the single experiments and of the LEP average are summarised for the differentcentre-of-mass energies in Table 27. Given the statistical precision of the single–W measurements,results are combined assuming uncorrelated systematic errors between experiments and using expectedstatistical errors.

Since winter 2000 conferences [129], the only update concerns the addition of one theoreticalprediction [130] to the single–W production cross section curve as a function of the centre-of-massenergy for the hadronic final state, based on the exact treatment of the evolution of the energy scalefor αem [131] in the 4f matrix element. This evolution is only approximated (at the level of 1–2%)by one of the codes previously used [132], and introduced by rescaling the cross section in one othercode [133]. The data at

√s = 183–202 GeV are compared with the theoretical predictions in Figure 11.

It should be remarked that all the theoretical predictions are scaled upward by 4% , since all codesimplement QED radiative corrections (mainly due to initial state radiation) using the wrong energyscale s in the ISR structure functions which is not adequate for this process dominated by t-channelexchange diagrams. The correction factor of 4% is derived in [134] from a comparison of grc4f [135]and SWAP [136] for different scales in the structure functions. Since this is only an approximation ofthe complete O(α) correction for the single W process, a 100% uncertainty is conservatively assignedto this 4% correction. This constitutes the major part of the theoretical uncertainty currently assignedto the predictions (5%, as discussed in [134]), which is becoming a limiting factor for the extractionof triple gauge couplings from the single W events [137].

44

√s Cross section (pb) χ2/d.o.f.

(GeV) ALEPH DELPHI L3 OPAL LEP182.67 0.40 ± 0.24 0.58+0.23

−0.20 0.50 ± 0.16 0.33/1188.63 0.31 ± 0.14 0.44+0.28

−0.25 0.52+0.14−0.13 0.53+0.13

−0.13 0.46 ± 0.07 1.71/3191.6 0.94 ± 0.44 0.01+0.19

−0.07 0.85+0.45−0.37 0.73 ± 0.26 1.70/2

195.5 0.45 ± 0.23 0.78+0.38−0.34 0.66+0.25

−0.23 0.60 ± 0.15 0.79/2199.5 0.82 ± 0.26 0.16+0.29

−0.17 0.34+0.23−0.20 0.45 ± 0.15 3.26/2

201.6 0.68 ± 0.35 0.55+0.47−0.40 1.09+0.42

−0.37 0.80 ± 0.22 1.03/2

Table 27: Single–W cross section for the hadronic decay of the W, from the four experiments and LEPcombined value for the five centre-of-mass energies. All numbers are preliminary.

0

0.25

0.5

0.75

1

160 170 180 190 200 210

Ecm [GeV]

σ(e+

e– → e

νW)

[pb]


±5.0% uncertainty

grc4fWPHACT IFL/α(t)WTO EFL

W → qq-’

Figure 11: Measurements of the single–W production cross section in the hadronic decay channel ofthe W boson compared to the predictions of WTO, WPHACT and grc4f. The shaded area representsa ±5% uncertainty on the predictions.

45

10 Measurement of the ZZ Production Cross Section

Updates with respect to last summer:All experiments have published results up to 189 GeV, new results at higher energies are available.

At centre-of-mass energies above twice the Z boson mass, the production of pairs of Z bosons ispossible. All experiments have published final results [138–141] on the Z–pair (NC02) productioncross section at

√s = 183 and 189 GeV. L3 [142] has updated its preliminary results for

√s = 192–202

GeV, whereas ALEPH [143], DELPHI [144] and OPAL [145] preliminary results are unchanged sincethe winter conferences. All experiments have provided preliminary results [94, 96–98] based on anintegrated luminosity of approximately 90 pb−1 collected in the year 2000 at centre-of-mass energiesbetween 200 and 208 GeV.

The combination of results is performed using the expected statistical error of each analysis, toavoid biases due to the limited number of events selected. The following components of the systematicerrors are considered correlated between experiments:

• ±5% uncertainty on the 2–fermion background rate.

• ±2% uncertainty on the WW background rate.

• ±10% uncertainty on the Z0e+e− background rate.

• the uncertainty on the b quark modelling.

The common error ranges from 0.01 to 0.03 pb.

The results of the single experiments and of the LEP average are summarised for the differentcentre-of-mass energies in Table 28. All results are preliminary, with the exception of those at

√s = 183

and 189 GeV. The results are shown in Figure 12 as a function of the LEP centre-of-mass energy andcompared to the YFSZZ [146] and ZZTO [147] predictions, which have an estimated uncertainty of±2% [134]. These results supersede those presented in [148] for

√s = 183–202 GeV. The data do not

show any significant deviation from the theoretical expectations.√s Cross section (pb) χ2/d.o.f.

(GeV) ALEPH DELPHI L3 OPAL LEP182.67 0.11+0.16∗

−0.12 0.38 ± 0.18∗ 0.31+0.17∗−0.15 0.12+0.20∗

−0.18 0.23 ± 0.08∗ 2.28/3188.63 0.67+0.14∗

−0.13 0.60 ± 0.15∗ 0.73+0.15∗−0.14 0.80+0.15∗

−0.14 0.70 ± 0.08∗ 0.92/3191.6 0.53+0.34

−0.27 0.55 ± 0.34 0.27+0.20−0.23 1.13+0.47

−0.41 0.60 ± 0.18 2.99/3195.5 0.69+0.23

−0.20 1.17 ± 0.28 1.11 ± 0.23 1.28+0.29−0.27 1.04 ± 0.12 3.43/3

199.5 0.70+0.22−0.20 1.08 ± 0.26 1.19 ± 0.24 1.01+0.26

−0.23 0.99 ± 0.13 2.31/3201.6 0.70+0.33

−0.28 0.87 ± 0.33 0.89+0.36−0.37 1.09+0.40

−0.35 0.88 ± 0.18 0.57/3204.9 0.86+0.29

−0.26 1.28 ± 0.34 0.56 ± 0.28 1.41+0.35−0.32 1.00 ± 0.15 4.74/3

206.7 0.51+0.35−0.28 1.11 ± 0.42 0.84 ± 0.39 0.30+0.37

−0.29 0.70 ± 0.21 2.13/3

Table 28: Z–pair production cross section from the four LEP experiments and combined values for theeight centre-of-mass energies. All results are preliminary with the exception of those indicated by ∗.A common systematic error of (0.01–0.03) pb is taken into account in the averaging procedure.

46

0

0.5

1

1.5

170 180 190 200

Ecm [GeV]

σZZ

NC

02 [p

b]


±2.0% uncertainty

ZZTO

YFSZZ

Figure 12: Measurements of the Z–pair production cross section compared to the predictions of YFSZZand ZZTO. The shaded area represent a ±2% uncertainty on the predictions.

47

11 Electroweak Gauge Boson Couplings

Updates with respect to last summer:This is a new section

The W-pair production process, e+e− → W+W−, involves the triple gauge boson vertices between theW+W− and the Z or the photon. Up-to the end of 1999, more than 7000 W-pair events were collectedby each experiment. Single W (eνW) and single photon (ννγ) production at LEP are sensitive tothe WWγ vertex. The measurement of these charged triple gauge boson couplings and the search forpossible anomalous contributions due to the effects of new physics beyond the Standard Model is oneof the principal physics goals at LEP-II [149].

At centre-of-mass energies exceeding twice the Z boson mass, pair production of Z bosons iskinematically allowed. Here, one searches for the possible existence of triple vertices involving onlyneutral electroweak gauge bosons. Such vertices could also contribute to Zγ production. In contrastto triple gauge boson vertices with two charged gauge bosons, purely neutral gauge boson vertices donot occur in the Standard Model of electroweak interactions.

Within the Standard Model, quartic electroweak gauge boson vertices with at least two chargedgauge bosons exist. In e+e− collisions at LEP-II centre-of-mass energies, the WWZγ and WWγγvertices contribute to WWγ and ννγγ production in s-channel and t-channel, respectively. The effectof the Standard Model quartic electroweak vertices is below the sensitivity of LEP-II. Thus, anomalousquartic vertices are searched for in the production of WWγ and ννγγ but also Zγγ final states.

In this note we present the combination of results on electroweak gauge boson couplings basedon the data collected by the four LEP experiments at LEP-II. Charged TGCs [150–153], neutralTGCs [154–159] as well as QGCs [160–162] are combined.

11.1 Charged Triple Gauge Boson Couplings

The parametrisation of the charged triple gauge boson vertices is described in References [149,163–168].The most general Lorentz invariant Lagrangian which describes the charged triple gauge boson inter-action contains fourteen independent complex couplings, seven describing the WWγ vertex and sevendescribing the WWZ vertex. Assuming electromagnetic gauge invariance as well as CP conservation,the number of independent TGCs reduces to six. One common set is {gz

1, κγ , κz, λγ , λz, gZ5 } where

gz1 = κγ = κz = 1 and λγ = λz = gZ

5 = 0 in the Standard Model. Except gZ5 , all these couplings also

conserve C and P separately.

The parameters proposed in [149] and used here are:

∆gz1 ≡ gz

1 − 1 , (23)∆κγ ≡ κγ − 1 , (24)λγ and gZ

5 (25)

with the gauge constraints:

∆κz ≡ κz − 1 = ∆gz1 −∆κγtan2 θW (26)

λz = λγ , (27)

48

where ∆ indicates the deviation of the respective coupling from its Standard Model value of unity,and θW is the weak mixing angle. The couplings are considered as real, with the imaginary parts fixedto zero.

11.2 Neutral Triple Gauge Boson Couplings

There are two possible anomalous triple vertices in the neutral sector. Each is parametrised bythe most general Lorentz and U(1)EM invariant Lagrangian plus Bose symmetry, as described inReferences [164,169].

The first vertex refers to anomalous Zγγ and ZZγ couplings which are accessible at LEP in theprocess e+e− → Zγ. The parametrisation contains eight couplings: hV

i with i = 1, ..., 4 and V = γ,Z.The superscript γ refers to Zγγ couplings and superscript Z refers to ZZγ couplings. The photon andthe Z boson in the final state are considered as on-shell particles, while the third boson at the triplevertex is off shell. The couplings hV

1 and hV2 are CP-odd while hV

3 and hV4 are CP-even.

The second vertex refers to anomalous ZZγ and ZZZ couplings which are accessible at LEP inthe process e+e− → ZZ. This anomalous vertex is parametrised in terms of four couplings: fV

i withi = 4, 5 and V = γ,Z. The superscript γ refers to ZZγ couplings and the superscript Z refers to ZZZcouplings. Both Z bosons in the final state are assumed to be on-shell, while the third boson at thetriple vertex is off-shell. The couplings fV

4 are CP-odd whereas fV5 are CP-even.

Note that the hVi and fV

i couplings are independent of each other. They are considered as real,with the imaginary parts fixed to zero. They vanish in the Standard Model at tree level.

11.3 Quartic Gauge Boson Couplings

Anomalous contributions to electroweak quartic vertices are considered in the framework of Refer-ences [170–172]. Three lowest-dimensional operators leading to quartic vertices not causing anomalousTGCs are considered. The corresponding couplings are a0/Λ2, ac/Λ2 and an/Λ2, where Λ representsthe energy scale of new physics. The couplings a0/Λ2 and ac/Λ2 parametrise variations in the WWγγand ZZγγ vertices, while an/Λ2 affects purely the WWZγ vertex. The couplings a0/Λ2 and ac/Λ2

conserve C and P, while the coupling an/Λ2 is CP violating. The production of WWγ depends onall three couplings. The production of ννγγ and Zγγ depends only on a0/Λ2 and ac/Λ2, the formerthrough the WWγγ vertex, the latter through the ZZγγ vertex. The couplings are considered as real,with the imaginary parts fixed to zero. They vanish in the Standard Model at tree level.

A more detailed description of QGCs has recently appeared [173], in which it is suggested that thecoupling a0/Λ2 and ac/Λ2 describing the anomalous WWγγ vertex should not be assumed to be thesame couplings as those describing the anomalous ZZγγ vertex. The experimental results on thesevertices are therefore also given separately.

11.4 Measurements

The results presented comprise measurements of all electroweak gauge boson couplings discussedabove. In most cases, the results are based on the data collected at LEP-II until the end of 1999 atcentre-of-mass energies up to 202 GeV. The experiments already combine different processes and final

49

states for each coupling. In each case, the individual references should be consulted for the details onthe data samples used.

For charged TGCs, the experimental analyses study different channels, typically the semileptonic,fully hadronic and fully leptonic W-pair decays [150–153]. Anomalous TGCs affect both the totalproduction cross section and the shape of the differential cross section as a function of the polarW− production angle. The relative contributions of each helicity state of the W bosons are alsochanged, which in turn affects the distributions of their decay products. The analyses presented byeach experiment make use of different combinations of each of these quantities. In general, however,all analyses use at least the expected variations of the total production cross section and the W−

production angle. Results from eνW and ννγ production are included by some experiments. SingleW production is particularly sensitive to κγ , thus providing information complementary to that fromW-pair production.

For neutral TGCs, the analyses mainly use measurements of the total cross sections of Zγ andZZ production, though the use of differential distributions increases the sensitivity. The individualreferences should be consulted concerning the details of the analyses for the hV

i couplings [154–156]and the fV

i couplings [157–159].9

For QGCs, the experiments investigate WWγ, ννγγ and Zγγ production. Besides the total crosssection, the sensitive variables analysed are photon energies (WWγ and Zγγ) and recoil masses (ννγγ).Again, the individual references should be consulted for the details in the determination of the QGCsa0/Λ2, ac/Λ2 and an/Λ2 [160–162].

11.5 Combination Procedure

The combination procedure is modified compared to previous LEP combinations of electroweak gaugeboson couplings [1,174] in order to treat systematic uncertainties correlated between the experiments.

Each experiment provides the negative log likelihood, −∆ logL, as a function of the couplingparameters (one, two, or three) to be combined. The single parameter analyses are performed fixingthe values of all other parameters to their Standard Model value. The two- and three-parameteranalyses are performed setting the remaining parameters to their Standard Model value. For thecharged TGCs, the gauge constraints listed in Section 11.1 are always enforced.

The −∆ logL functions from each experiment include statistical as well as all systematic un-certainties considered as uncorrelated between experiments. For both single- and multi-parametercombinations, the individual −∆ logL functions are added. It is necessary to use the −∆ logL func-tions directly for the combination as in some cases they are not parabolic, so that it is not possible tocombine the results correctly by simply taking weighted averages of the measurements.

The main contributions to the systematic uncertainties uncorrelated between experiments arisefrom detector effects, backgrounds in the selected samples, limited Monte Carlo statistics and thefitting methods. Their importance varies for each experiment and the individual references should beconsulted for details.

The systematic uncertainties arising from the following sources are treated as correlated between9For quoting results on the fV

i couplings, DELPHI [157] uses a convention for the sign of fVi opposite to that used by

L3 [158] and OPAL [159]. For DELPHI and combined LEP results presented in this note, the DELPHI fVi measurements

are modified to conform to the same sign convention as used by L3 and OPAL.

50

the experiments: the uncertainty on the theoretical cross section prediction (W+W−, eνW and Zγproduction), fragmentation effects (W+W− production), and Bose-Einstein correlations and colour-reconnection effects (fully hadronic W-pair decays). While the correlated systematic uncertainties aresmall in Zγ production, they are sizeable for charged TGCs. In particular, the uncertainty on thetheoretical cross section prediction for the eνW process causes a large reduction in the sensitivity ofthis process to charged TGCs. For ZZ production, the uncertainty on the theoretical cross sectionprediction is small compared to the statistical accuracy and is neglected. For other sources of correlatedsystematic uncertainties, such as those arising from the LEP beam energy or the W mass, the effectof their correlation on the combination result is negligible.

The correlated systematic uncertainties are applied by scaling the likelihood functions by thesquared ratio of the statistical and uncorrelated systematic uncertainty over the total uncertaintyincluding all correlated uncertainties. This procedure to treat correlations is an approximation in thecase where the log-likelihood function is not parabolic.

The one standard deviation uncertainties (68% confidence level) are obtained by taking the couplingvalues where −∆ logL = +0.5 above the minimum. The 95% confidence level (C.L.) limits are givenby the coupling values where −∆ logL = +1.92 above the minimum. These cut-off values are usedfor obtaining the results of both single- and multi-parameter analyses reported here. Note that inthe case of the neutral TGCs and the QGCs, double minima structures appear in the negative log-likelihood curves. For multi-parameter analyses, the 68% C.L. contour curves for any pair of couplingsare obtained by requiring −∆ logL = +1.15, while for the 95% C.L. contour curves −∆ logL = +3.0is required.

11.6 Results

We present below results on combination of the four LEP experiments on the various electroweak gaugeboson couplings. The results quoted individually for each experiment, summarised in Appendix B arecalculated using the method described in Section 11.5. Thus, they sometimes differ slightly from thosereported in the individual references.

11.6.1 Charged Triple Gauge Boson Couplings

Results of the combination of charged triple gauge boson couplings are given in Tables 29, 30 and 31 forthe single, two and three-parameter analyses respectively. Individual −∆ logL curves and their sumare shown in Figure 13 for the single-parameter analysis. The 68% C.L. and 95% C.L. contours curvesresulting from the combinations of the two-dimensional likelihood curves are shown in Figure 14.

51

Parameter 68% C.L. 95% C.L.

gZ5 +0.05+0.16

−0.17 [−0.28, +0.36]

∆gz1 −0.025+0.026

−0.026 [−0.074, +0.028]

∆κγ −0.002+0.067−0.065 [−0.13, +0.13]

λγ −0.036+0.028−0.027 [−0.089, +0.020]

Table 29: Single-parameter analyses : the central values, 68% C.L. uncertainties and 95% C.L. intervals(−∆ logL = 0.5, 1.92) obtained combining the results from the four LEP experiments. In each casethe parameter listed is varied while the remaining ones are fixed to their Standard Model value. Bothstatistical and systematic uncertainties are included.

Parameter 68% C.L. 95% C.L. Correlations

∆gz1 −0.038+0.041

−0.034 [−0.10, +0.03] 1.00 −0.43

∆κγ +0.053+0.074−0.089 [−0.10, +0.21] −0.43 1.00

∆gz1 +0.006+0.035

−0.039 [−0.072, +0.076] 1.00 −0.70

λγ −0.048+0.043−0.036 [−0.12, +0.03] −0.70 1.00

λγ −0.046+0.041−0.031 [−0.11, +0.02] 1.00 −0.19

∆κγ +0.037+0.060−0.069 [−0.09, +0.16] −0.19 1.00

Table 30: Two-parameter analyses : the central values, 68% C.L. uncertainties and 95% C.L. intervals(−∆ logL = 0.5, 1.92) obtained combining the results from the four LEP experiments. In each casethe two parameters listed are varied while the remaining one is fixed to its Standard Model value.Both statistical and systematic uncertainties are included. Since the shape of the log-likelihood isnot parabolic, there is some ambiguity in the definition of the correlation coefficients and the valuesquoted here are approximate.

Parameter 68% C.L. 95% C.L. Correlations

∆gz1 +0.023+0.039

−0.041 [−0.06, +0.10] 1.00 −0.20 −0.66

∆κγ +0.027+0.073−0.072 [−0.11, +0.18] −0.20 1.00 −0.15

λγ −0.067+0.044−0.037 [−0.14, +0.02] −0.66 −0.15 1.00

Table 31: Three-parameter analyses : the central values, 68% C.L. uncertainties and 95% C.L. inter-vals (−∆ logL = 0.5, 1.92) obtained combining the results from ALEPH, L3 and OPAL. All threeparameters listed are varied. Both statistical and systematic uncertainties are included. Since theshape of the log-likelihood is not parabolic, there is some ambiguity in the definition of the correlationcoefficients and the values quoted here are approximate.

52

11.6.2 Neutral Triple Gauge Boson Couplings in Zγ Production

Results of the combination of neutral triple gauge boson couplings in Zγ production are given inTables 32 and 33 for the single, two-parameter analyses respectively. The 68% C.L. and 95% C.L.contours curves resulting from the combinations of the two-dimensional likelihood curves are shownin Figure 15.

Parameter 95% C.L.

hγ1 [−0.10, +0.03]

hγ2 [−0.036, +0.061]

hγ3 [−0.075, −0.004]

hγ4 [+0.005, +0.056]

hZ1 [−0.13, +0.04]

hZ2 [−0.041, +0.080]

hZ3 [−0.16, +0.07]

hZ4 [−0.04, +0.10]

Table 32: Single-parameter analyses : The 95% C.L. intervals (−∆ logL = 1.92) obtained combiningthe results from DELPHI, L3 and OPAL. In each case the parameter listed is varied while the remain-ing ones are fixed to their Standard Model value. Both statistical and systematic uncertainties areincluded.

Parameter 95% C.L. Correlations

hγ1 [−0.21, +0.10] 1.00 +0.88

hγ2 [−0.11, +0.10] +0.88 1.00

hγ3 [−0.20, +0.13] 1.00 +0.98

hγ4 [−0.11, +0.12] +0.98 1.00

hZ1 [−0.44, +0.20] 1.00 +0.96

hZ2 [−0.28, +0.16] +0.96 1.00

hZ3 [−0.38, +0.35] 1.00 +0.95

hZ4 [−0.21, +0.26] +0.95 1.00

Table 33: Two-parameter analyses : The 95% C.L. intervals (−∆ logL = 1.92) obtained combiningthe results from DELPHI and L3. In each case the two parameters listed are varied while the re-maining ones are fixed to their Standard Model value. Both statistical and systematic uncertaintiesare included. Since the shape of the log-likelihood is not parabolic, there is some ambiguity in thedefinition of the correlation coefficients and the values quoted here are approximate.

53

11.6.3 Neutral Triple Gauge Boson Couplings in ZZ Production

Results of the combination of neutral triple gauge boson couplings in ZZ production are given inTables 34 and 35 for the single, two-parameter analyses respectively. The 68% C.L. and 95% C.L.contours curves resulting from the combinations of the two-dimensional likelihood curves are shownin Figure 16.

Parameter 95% C.L.

fγ4 [−0.41, +0.39]

fZ4 [−0.66, +0.68]

fγ5 [−0.89, +0.84]

fZ5 [−1.1, +0.5]

Table 34: Single-parameter analyses : The 95% C.L. intervals (−∆ logL = 1.92) obtained combiningthe results from DELPHI, L3 and OPAL. In each case the parameter listed is varied while the remain-ing ones are fixed to their Standard Model value. Both statistical and systematic uncertainties areincluded.

Parameter 95% C.L. Correlations

fγ4 [−0.40, +0.38] 1.00 +0.88

fZ4 [−0.66, +0.68] +0.88 1.00

fγ5 [−0.88, +0.86] 1.00 −0.52

fZ5 [−1.1, +0.7] −0.52 1.00

Table 35: Two-parameter analyses : The 95% C.L. intervals (−∆ logL = 1.92) obtained combiningthe results from DELPHI, L3 and OPAL. In each case the two parameters listed are varied while theremaining ones are fixed to their Standard Model value. Both statistical and systematic uncertaintiesare included. Since the shape of the log-likelihood is not parabolic, there is some ambiguity in thedefinition of the correlation coefficients and the values quoted here are approximate.

54

11.6.4 Quartic Gauge Boson Couplings

The individual−∆ logL curves and their sum are shown in Figure 17 for the WWγ and ννγγ channels.The results of the combination are given in Table 36.

Parameter 95% C.L.

[GeV−2]

a0/Λ2 [−0.037, +0.036]

ac/Λ2 [−0.077, +0.095]

an/Λ2 [−0.45, +0.41]

a0/Λ2 [−0.0048, +0.0056]

ac/Λ2 [−0.0052, +0.0099]

Table 36: The 95% C.L. intervals (−∆ logL = 1.92) obtained combining the results from ALEPH,L3 and OPAL. In each case the parameter listed is varied while the remaining ones are fixed to theirStandard Model value. Both statistical and systematic uncertainties are included. Top: WWγ andννγγ. Bottom: Zγγ.

11.7 Conclusions

The existence of triple gauge boson couplings among the electroweak gauge bosons is experimentallyverified. No significant deviation from the Standard Model predictions is seen for any of the electroweakgauge boson couplings investigated. As an example, these data allow the Kaluza-Klein theory [175],in which κγ = −2, to be excluded [176].

55

0

0.5

1

1.5

2

2.5

3

-0.4 -0.2 0 0.2 0.4∆kγ

-∆lo

g l

0

0.5

1

1.5

2

2.5

3

-0.2 -0.1 0 0.1 0.2∆g

1

Z

-∆lo

g l

0

0.5

1

1.5

2

2.5

3

-0.2 -0.1 0 0.1 0.2λγ

-∆lo

g l

ALEPH + DELPHI +L3 + OPAL

∆kγ= -0.002

∆g1

Z= -0.025

λγ = -0.036

+ 0.067_ 0.065

+ 0.026_ 0.026

+ 0.028_ 0.027

Preliminary

Figure 13: The −∆ logL curves of the four experiments (dashed lines) and the LEP combined curve(solid line) for the three charged TGCs ∆gz

1, ∆κγ and λγ . In each case, the minimal value is subtracted.

56

∆g1Z

∆κγ

68% CL95% CL SM

PreliminaryLEP

-0.3

-0.15

0

0.15

0.3

-0.3 -0.15 0 0.15 0.3

∆g1Z

λ γ

68% CL95% CL SM

PreliminaryLEP

-0.3

-0.15

0

0.15

0.3

-0.3 -0.15 0 0.15 0.3

λγ

∆κγ

68% CL95% CL SM

PreliminaryLEP

-0.3

-0.15

0

0.15

0.3

-0.3 -0.15 0 0.15 0.3

Figure 14: Contour curves of 68% C.L. and 95% C.L. in the planes (∆gz1,∆κγ), (∆gz

1, λγ), (λγ ,∆κγ)showing the LEP combined results.

57

h1γ

h 2γ

68% CL95% CL SM

PreliminaryLEP

-0.3

-0.15

0

0.15

0.3

-0.3 -0.15 0 0.15 0.3

h3γ

h 4γ

68% CL95% CL SM

PreliminaryLEP

-0.3

-0.15

0

0.15

0.3

-0.3 -0.15 0 0.15 0.3

h1Z

h 2Z

68% CL95% CL SM

PreliminaryLEP

-0.6

-0.3

0

0.3

0.6

-0.6 -0.3 0 0.3 0.6

h3Z

h 4Z

68% CL95% CL SM

PreliminaryLEP

-0.6

-0.3

0

0.3

0.6

-0.6 -0.3 0 0.3 0.6

Figure 15: Contour curves of 68% C.L. and 95% C.L. in the planes (hγ1 , h

γ2), (hγ

3 , hγ4), (hZ

1 , hZ2 ) and

(hZ3 , h

Z4 ) showing the LEP combined results.

58

f4γ

f 4Z

68% CL95% CL SM

PreliminaryLEP

-1

-0.5

0

0.5

1

-1 -0.5 0 0.5 1

f5γ

f 5Z

68% CL95% CL SM

PreliminaryLEP

-2

-1

0

1

2

-2 -1 0 1 2

Figure 16: Contour curves of 68% C.L. and 95% C.L. in the planes (fγ4 , f

Z4 ) and (fγ

5 , fZ5 ) showing the

LEP combined results.

59

0

0.5

1

1.5

2

2.5

3

-0.1 -0.05 0 0.05 0.10

0.5

1

1.5

2

2.5

3

-0.2 -0.1 0 0.1 0.2

0

0.5

1

1.5

2

2.5

3

-0.5 -0.25 0 0.25 0.5

a0/Λ2

-∆lo

g l

ac/Λ2

-∆lo

g l

an/Λ2

-∆lo

g l

Preliminary ALEPH +L3+OPAL

a0/Λ2, ac/Λ

2: ννγγ+WWγ

an/Λ2: WW γ

Figure 17: The −∆ logL curves of ALEPH, L3 and OPAL, and the LEP combined curve for the QGCsderived from WWγ and ννγγ production. In each case, the minimal value is subtracted.

60

12 Interpretation of Results

Updates with respect to last summer:Results on sin2θlept

eff from asymmetries are averaged for leptonic and hadronic channels separately. Anew fit for the top mass is performed where all data except the direct measurement of mt is included.

12.1 Number of Neutrino Species

An important aspect of our measurement concerns the information related to Z decays into invisiblechannels. Using the results of Table 3, the ratio of the Z decay width into invisible particles and theleptonic decay width is determined:

Γinv/Γ`` = 5.942 ± 0.016 . (28)

The Standard Model value for the ratio of the partial widths to neutrinos and charged leptons is:

(Γνν/Γ``)SM = 1.9912 ± 0.0012 . (29)

The central value is evaluated for mZ = 91.1875 GeV and the error quoted accounts for a variation ofmt in the range mt = 174.3±5.1 GeV and a variation of mH in the range 100 GeV ≤ mH ≤ 1000 GeV.The number of light neutrino species is given by the ratio of the two expressions listed above:

Nν = 2.9841 ± 0.0083, (30)

which is 2 standard deviations below the expected value of 3.

Alternatively, one can assume 3 neutrino species and determine the width from additional invisibledecays of the Z. This yields

∆Γinv = −2.7± 1.6 MeV. (31)

The measured total width is below the Standard Model expectation. If a conservative approach istaken to limit the result to only positive values of ∆Γinv, then the 95% CL upper limit on additionalinvisible decays of the Z is

∆Γinv < 2.0 MeV. (32)

The uncertainties on Nν and ∆Γinv are dominated by the theoretical error on the luminosity. Theseresults have therefore improved due to the improved theoretical calculations on Bhabha scattering [14].

12.2 The Coupling Parameters Af

The coupling parameters Af are defined in terms of the effective vector and axial-vector neutral currentcouplings of fermions (Equation (4)). The LEP measurements of the forward-backward asymmetriesof charged leptons (Section 2) and b and c quarks (Section 5) determine the products A0, f

FB = 34AeAf

(Equation (3)). The LEP measurements of the τ polarisation (Section 3), Pτ (cos θ), determine Aτ

and Ae separately (Equation (6)).

61

A` Cumulative Average χ2/d.o.f.

A0, `FB 0.1512 ± 0.0042

Pτ (cos θ) 0.1464 ± 0.0032 0.1482 ± 0.0025 0.8/1A` (SLD) 0.1513 ± 0.0021 0.1500 ± 0.0016 1.7/2

Table 37: Determination of the leptonic coupling parameter A` assuming lepton universality. Thesecond column lists the A` values derived from the quantities listed in the first column. The thirdcolumn contains the cumulative averages of these A` results. The averages are derived assuming nocorrelations between the measurements. The χ2 per degree of freedom for the cumulative averages isgiven in the last column.

LEP SLD LEP+SLD Standard(A` = 0.1482 ± 0.0025) (A` = 0.1500 ± 0.0016) Model fit

Ab 0.890 ± 0.024 0.922 ± 0.023 0.898 ± 0.015 0.935Ac 0.619 ± 0.035 0.631 ± 0.026 0.623 ± 0.020 0.668

Table 38: Determination of the quark coupling parameters Ab and Ac from LEP data alone (using theLEP average for A`), from SLD data alone, and from LEP+SLD data (using the LEP+SLD averagefor A`) assuming lepton universality.

Table 37 shows the results for the leptonic coupling parameter A` from the LEP and SLD mea-surements, assuming lepton universality.

Using the measurements of A` one can extract Ab and Ac from the LEP measurements of the band c quark asymmetries. The SLD measurements of the left-right forward-backward asymmetriesfor b and c quarks are direct determinations of Ab and Ac. Table 38 shows the results on the quarkcoupling parameters Ab and Ac derived from LEP or SLD measurements separately (Equations 17and 18) and from the combination of LEP+SLD measurements (Equation 18). The LEP extractedvalues of Ab and Ac in excellent agreement with the SLD measurements, and in reasonable agreementwith the Standard Model predictions (0.935 and 0.668, respectively, essentially independent of mt andmH). The combination of LEP and SLD of Ab and Ac are 2.5 and 2.3 sigma below the StandardModel respectively. This is mainly because the Ab value deduced from the measured A0, b

FB and thecombined A` is low compared to both the Standard Model and the direct measurement of Ab, thiscan also be seen in Figure 18.

12.3 The Effective Vector and Axial-Vector Coupling Constants

The partial widths of the Z into leptons and the lepton forward-backward asymmetries (Section 2), theτ polarisation and the τ polarisation asymmetry (Section 3) are combined to determine the effectivevector and axial-vector couplings for e, µ and τ . The asymmetries (Equations (3) and (6)) determinethe ratio gV `/gA` (Equation (4)), while the leptonic partial widths determine the sum of the squaresof the couplings:

Γ`` =GFm

3Z

6π√

2(g2

V ` + g2A`)(1 + δQED

` ) , (33)

62

0.7

0.8

0.9

1

0.13 0.14 0.15 0.16 0.17

Al

Ab

Preliminary

SM

0.5

0.6

0.7

0.8

0.13 0.14 0.15 0.16 0.17

Al

Ac

Preliminary

SM

Figure 18: The measurements of the combined LEP+SLD A` (vertical band), SLD Ab,Ac (horizontalbands) and LEP A0, b

FB ,A0, cFB (diagonal bands), compared to the Standard Model expectations (arrows).

The arrow pointing to the left shows the variation in the SM prediction for mH in the range 300+700−187

GeV, and the arrow pointing to the right for mt in the range 174.3 ± 5.1 GeV. Although the A0, bFB

measurements prefer a high Higgs mass, the Standard Model fit to the full set of measurements prefersa low Higgs mass, for example because of the influence of A`.

where δQED` = 3q2`α(m2

Z)/(4π) accounts for final state photonic corrections. Corrections due to leptonmasses, neglected in Equation 33, are taken into account for the results presented below.

The averaged results for the effective lepton couplings are given in Table 39 for both the LEP dataalone as well as for the LEP and SLD measurements. Figure 19 shows the 68% probability contoursin the gA`-gV ` plane for the individual lepton species from the LEP data. The signs of gA` and gV ` arebased on the convention gAe < 0. With this convention the signs of the couplings of all charged leptonsfollow from LEP data alone. For comparison, the gV `-gA` relation following from the measurement ofA` from SLD [177] is indicated as a band in the gA`-gV `-plane of Figure 19. The measured ratios ofthe e, µ and τ couplings provide a test of lepton universality and are shown in Table 39. All valuesare consistent with lepton universality. The combined results assuming universality are also given inthe table and are shown as a solid contour in Figure 19.

The neutrino couplings to the Z can be derived from the measured value of the invisible width of theZ, Γinv (see Table 4), attributing it exclusively to the decay into three identical neutrino generations(Γinv = 3Γνν) and assuming gAν ≡ gV ν ≡ gν . The relative sign of gν is chosen to be in agreement withneutrino scattering data [178], resulting in gν = +0.50068 ± 0.00075.

63

-0.043

-0.039

-0.035

-0.031

-0.503 -0.502 -0.501 -0.5

gAl

g Vl

Preliminary

68% CL

Combinede+e−

µ+µ−

τ+τ−

Al (SLD)

mt

mH

Figure 19: Contours of 68% probability in the gV `-gA` plane from LEP measurements. Also shown isthe one standard deviation band resulting from the A` measurement of SLD. The solid contour resultsfrom a fit to the LEP and SLD results assuming lepton universality. The shaded region correspondsto the Standard Model prediction for mt = 174.3 ± 5.1 GeV and mH = 300+700

−187 GeV. The arrowspoint in the direction of increasing values of mt and mH.

64

Without Lepton Universality:LEP LEP+SLD

gV e −0.0378 ± 0.0011 −0.03809 ± 0.00047gV µ −0.0376 ± 0.0031 −0.0360 ± 0.0024gV τ −0.0367 ± 0.0010 −0.0364 ± 0.0010gAe −0.50112 ± 0.00035 −0.50111 ± 0.00035gAµ −0.50115 ± 0.00056 −0.50120 ± 0.00054gAτ −0.50205 ± 0.00064 −0.50204 ± 0.00064

Ratios of couplings:LEP LEP+SLD

gV µ/gV e 0.995 ± 0.093 0.962 ± 0.062gV τ/gV e 0.972 ± 0.039 0.958 ± 0.029gAµ/gAe 1.0001 ± 0.0014 1.0002 ± 0.0014gAτ/gAe 1.0018 ± 0.0015 1.0019 ± 0.0015

With Lepton Universality:LEP LEP+SLD

gV ` −0.03734 ± 0.00065 −0.03782 ± 0.00040gA` −0.50126 ± 0.00026 −0.50123 ± 0.00026gν +0.50068 ± 0.00075 +0.50068 ± 0.00075

Table 39: Results for the effective vector and axial-vector couplings derived from the combined LEPdata without and with the assumption of lepton universality. For the right column the SLD measure-ments of A0

LR, Ae, Aµ and Aτ are also included.

12.4 The Effective Electroweak Mixing Angle sin2θlepteff

The asymmetry measurements from LEP and SLD can be combined into a single observable, theeffective electroweak mixing angle, sin2θlept

eff , defined as:

sin2θlepteff ≡ 1

4

(1− gV `

gA`

), (34)

without making strong model-specific assumptions.

For a combined average of sin2θlepteff from A0, `

FB, Aτ andAe only the assumption of lepton universality,already inherent in the definition of sin2θlept

eff , is needed. Also the value derived from the measurementsof A` from SLD is given. We can also include the hadronic forward-backward asymmetries if we assumethe quark couplings to be given by the Standard Model. This is justified within the Standard Modelas the hadronic asymmetries A0, b

FB and A0, cFB have a reduced sensitivity to corrections particular to the

quark vertex. The results of these determinations of sin2θlepteff and their combination are shown in

Table 40 and in Figure 20. The combinations based on leptonic and hadronic results show a differenceof 3.0 sigma, mainly caused by the two most precise measurements of sin2θlept

eff , A0LR (SLD) and A0, b

FB

(LEP). This is the same effect as discussed in section 12.2 and shown in Figure 18.

65

10 2

10 3

0.23 0.232 0.234

Preliminary

sin2θlept

eff

mH [

GeV

]

χ2/d.o.f.: 11.9 / 6

A0,l

fb 0.23099 ± 0.00053

Aτ 0.23192 ± 0.00053

Ae 0.23117 ± 0.00061

Al(SLD) 0.23098 ± 0.00026

A0,b

fb 0.23225 ± 0.00036

A0,c

fb 0.23262 ± 0.00082

<Qfb> 0.2321 ± 0.0010

Average 0.23146 ± 0.00017

∆αhad= 0.02804 ± 0.00065∆α(5)

αs= 0.119 ± 0.002mt= 174.3 ± 5.1 GeV

Figure 20: Comparison of several determinations of sin2θlepteff from asymmetries. In the average, the

small correlation between A0, bFB and A0, c

FB is included. Also shown is the prediction of the StandardModel as a function of mH. The width of the Standard Model band is due to the uncertainties in∆α(5)

had(m2Z) (see Section 12.5), αs(m2

Z) and mt. The total width of the band is the linear sum of theseeffects.

66

sin2θlepteff Average by Group Cumulative

of Observations Average χ2/d.o.f.

A0, `FB 0.23099 ± 0.00053

Aτ 0.23192 ± 0.00053

Ae 0.23117 ± 0.00061 0.23138 ± 0.00032 0.23138 ± 0.00032 1.7/2

A` (SLD) 0.23098 ± 0.00026 0.23098 ± 0.00026 0.23114 ± 0.00020 2.6/3

A0, bFB 0.23225 ± 0.00036

A0, cFB 0.23262 ± 0.00082

〈QFB〉 0.2321 ± 0.0010 0.23228 ± 0.00032 0.23146 ± 0.00017 11.9/6

Table 40: Determinations of sin2θlepteff from asymmetries. The second column lists the sin2θlept

eff valuesderived from the quantities listed in the first column. The third column contains the averages of thesenumbers by groups of observations, where the groups are separated by the horizontal lines. The fourthcolumn shows the cumulative averages. The χ2 per degree of freedom for the cumulative averages isalso given. The averages have been performed including the small correlation between A0, b

FB and A0, cFB.

67

12.5 Constraints on the Standard Model

The precise electroweak measurements performed at LEP and SLC and elsewhere can be used tocheck the validity of the Standard Model and, within its framework, to infer valuable informationabout its fundamental parameters. The accuracy of the measurements makes them sensitive to themass of the top quark mt, and to the mass of the Higgs boson mH through loop corrections. Whilethe leading mt dependence is quadratic, the leading mH dependence is logarithmic. Therefore, theinferred constraints on mH are not very strong.

The LEP and SLD [177] measurements used are summarised in Table 41 together with the resultsof the Standard Model fit. Also shown are the results of measurements of mW from UA2 [179],CDF [180,181], and DØ [182]10, measurements of the top quark mass by CDF [184] and DØ [185]11,and measurements of the neutrino-nucleon neutral to charged current ratios from CCFR [187] andNuTeV [188]. Although these latter results are quoted in terms of sin2 θW = 1 −m2

W/m2Z, radiative

corrections result in small mt and mH dependences12 that are included in the fit. In addition, the valueof the electromagnetic coupling constant α(m2

Z), which is used in the fits, is shown. An additionalinput parameter, not shown in the table, is the Fermi constant GF , determined from the µ lifetime,GF = (1.16637 ± 0.00001) × 10−5GeV−2 [189]. The relative error of GF is comparable to that of mZ;both errors have negligible effects in the fit results.

Detailed studies of the theoretical uncertainties in the Standard Model predictions due to missinghigher-order electroweak corrections and their interplay with QCD corrections have been carried out inthe working group on ‘Precision calculations for the Z resonance’ [192]. Theoretical uncertainties areevaluated by comparing different but, within our present knowledge, equivalent treatments of aspectssuch as resummation techniques, momentum transfer scales for vertex corrections and factorisationschemes. The effects of these theoretical uncertainties have been reduced by the inclusion of higher-order corrections [193, 194] in the electroweak libraries [195]. The use of the new QCD corrections[194] increases the value of αs(m2

Z) by 0.001, as expected. The effects of missing higher-order QCDcorrections on αs(m2

Z) covers missing higher-order electroweak corrections and uncertainties in theinterplay of electroweak and QCD corrections and is estimated to be about 0.002 [196]. A discussionof theoretical uncertainties in the determination of αs can be found in References 192 and 196. Forthe moment, the determination of the size of remaining theoretical uncertainties is still under study.All theoretical errors discussed in this paragraph are neglected for the results presented in Table 42.

At present the impact of theoretical uncertainties on the determination of SM parameters fromthe precise electroweak measurements is small compared with the error due to the uncertainty in thevalue of α(m2

Z). The uncertainty in α(m2Z) arises from the contribution of light quarks to the photon

vacuum polarisation (∆α(5)had(m2

Z)):

α(m2Z) =

α(0)

1−∆α`(m2Z)−∆α(5)

had(m2Z)−∆αtop(m2

Z). (35)

The top contribution depends on the mass of the top quark, and is therefore determined insidethe electroweak libraries [195]. The leptonic contribution is calculated to third order [191] to be0.031498. For the hadronic contribution, we use the value 0.02804 ± 0.00065 [190], which results in1/α(5)(m2

Z) = 128.878 ± 0.090. This uncertainty causes an error of 0.00023 on the Standard Model

10See Reference 183 for a combination of these mW measurements.11See Reference 186 for a combination of these mt measurements.12The formula used is δ sin2 θW = −0.00142

m2t−(175GeV)2

(100GeV)2+ 0.00048 ln( mH

150GeV). See Reference 188 for details.

68

Measurement with Systematic Standard PullTotal Error Error Model fit

∆α(5)had(m2

Z) [190,191] 0.02804 ± 0.00065 0.00064 0.02804 0.0

a) LEPline-shape andlepton asymmetries:mZ [GeV] 91.1875 ± 0.0021 (a)0.0017 91.1874 0.0ΓZ [GeV] 2.4952 ± 0.0023 (a)0.0012 2.4962 −0.4σ0

h [nb] 41.540 ± 0.037 (b)0.028 41.480 1.6R` 20.767 ± 0.025 (b)0.007 20.740 1.1A0, `

FB 0.0171 ± 0.0010 (b)0.0003 0.0164 0.8+ correlation matrix Table 3

τ polarisation:Aτ 0.1439 ± 0.0042 0.0026 0.1480 −1.0Ae 0.1498 ± 0.0048 0.0009 0.1480 0.4

qq charge asymmetry:sin2θlept

eff (〈QFB〉) 0.2321 ± 0.0010 0.0008 0.23140 0.7

mW [GeV] 80.427 ± 0.046 0.035 80.402 0.5

b) SLD [177]sin2θlept

eff (A`) 0.23098 ± 0.00026 0.00018 0.23140 −1.6

c) LEP and SLD Heavy FlavourR0

b 0.21653 ± 0.00069 0.00053 0.21578 1.1R0

c 0.1709 ± 0.0034 0.0022 0.1723 −0.4A0, b

FB 0.0990 ± 0.0020 0.0009 0.1038 −2.4A0, c

FB 0.0689 ± 0.0035 0.0017 0.0742 −1.5Ab 0.922 ± 0.023 0.016 0.935 −0.6Ac 0.631 ± 0.026 0.016 0.668 −1.4+ correlation matrix Table 10

d) pp and νNmW [GeV] (pp [183]) 80.452 ± 0.062 0.050 80.402 0.81−m2

W/m2Z (νN [187,188]) 0.2255 ± 0.0021 0.0010 0.2226 1.2

mt [GeV] (pp [186]) 174.3 ± 5.1 4.0 174.3 0.0

Table 41: Summary of measurements included in the combined analysis of Standard Model param-eters. Section a) summarises LEP averages, Section b) SLD results (sin2θlept

eff includes ALR and thepolarised lepton asymmetries), Section c) the LEP and SLD heavy flavour results and Section d)electroweak measurements from pp colliders and νN scattering. The total errors in column 2 includethe systematic errors listed in column 3. Although the systematic errors include both correlated anduncorrelated sources, the determination of the systematic part of each error is approximate. TheStandard Model results in column 4 and the pulls (difference between measurement and fit in unitsof the total measurement error) in column 5 are derived from the Standard Model fit including alldata (Table 42, column 5) with the Higgs mass treated as a free parameter.(a)The systematic errors on mZ and ΓZ contain the errors arising from the uncertainties in the LEP energyonly.(b)Only common systematic errors are indicated.

69

prediction of sin2θlepteff , an error of 1 GeV on mt, and 0.2 on log(mH), which are included in the

results. The effect on the Standard Model prediction for Γ`` is negligible. The αs(m2Z) values for the

Standard Model fits presented in this Section are stable against a variation of α(m2Z) in the interval

quoted.

At the ICHEP 2000 conference in Osaka, the BES collaboration reported on new preliminarymeasurements of the hadronic cross section in electron-positron collisions at 2 to 5 GeV centre-of-mass energy [197]. With these data, the hadronic vacuum polarisation is determined with improvedprecision: ∆α(5)

had(m2Z) = 0.02755 ± 0.00046 [198]. To show the effects of the uncertainty of α(m2

Z),we also use this evaluation of the hadronic vacuum polarisation. There are also several evaluations of∆α(5)

had(m2Z) [190, 199–207] which are more theory driven. The most recent of these (Reference 206)

also includes the new preliminary results from BES. All these evaluations obtain values for ∆α(5)had(m

2Z)

consistently lower than the old value of 0.02804 ± 0.00065.

Figure 21 shows a comparison of the leptonic partial width from LEP (Table 4) and the effectiveelectroweak mixing angle from asymmetries measured at LEP and SLD (Table 40), with the Stan-dard Model. Good agreement with the Standard Model prediction is observed. The point with thearrow shows the prediction if among the electroweak radiative corrections only the photon vacuumpolarisation is included, which shows an example of evidence that LEP+SLD data are sensitive toelectroweak corrections. Note that the error due to the uncertainty on α(m2

Z) (shown as the length ofthe arrow) is larger than the experimental error on sin2θlept

eff from LEP and SLD. This underlines thegrowing importance of a precise measurement of σ(e+e− → hadrons) at low centre-of-mass energies.

Of the measurements given in Table 41, R` is one of the most sensitive to QCD corrections. FormZ = 91.1875 GeV, and imposing mt = 174.3±5.1 GeV as a constraint, αs = 0.123±0.004 is obtained.Alternatively, σ0

` (see Table 4) which has higher sensitivity to QCD corrections and less dependenceon mH yields : αs = 0.118 ± 0.003. Typical errors arising from variation mH are of the order of0.001, somewhat smaller for σ0

` . These results are in very good agreement with the world average(αs(m2

Z) = 0.119 ± 0.002 [107]).

To test the agreement between the LEP data and the Standard Model, a fit to the data (includingthe LEP-II mW determination) leaving the top quark mass and the Higgs mass as free parametersis performed. The result is shown in Table 42, column 1. This fit shows that the LEP data prefera light top quark and a light Higgs boson, albeit with very large errors. The strongly asymmetricerrors on mH are due to the fact that to first order, the radiative corrections in the Standard Modelare proportional to log(mH). The correlation between the top quark mass and the Higgs mass is 0.63(see Figure 22).

The data can also be used within the Standard Model to determine the top quark and W massesindirectly, which can be compared to the direct measurements performed at the pp colliders and LEP.In the second fit, all the results in Table 41, except the LEP-II and pp colliders mW and mt resultsare used. The results are shown in column 2 of Table 42. The indirect measurements of mW and mt

from this data sample are shown in Figure 23, compared with the direct measurements. Also shownis the Standard Model predictions for Higgs masses between 113 and 1000 GeV. As can be seen inthe figure, the indirect and direct measurements of mW and mt are in good agreement, and both setsprefer a low Higgs mass.

For the third fit, the direct mt measurements is used to obtain the best indirect determination ofmW. The result is shown in column 3 of Table 42. Also here, the indirect determination of W bosonmass 80.386± 0.025 GeV is in excellent agreement with the combination of direct measurements fromLEP and pp colliders [183] of mW = 80.436 ± 0.037 GeV.

70

For the next fit, (column 4 of Table 42), the direct mW measurements from LEP and pp collidersare included to obtain mt = 174.4+9.3

−6.7 GeV, in good agreement with the direct measurement ofmt = 174.3 ± 5.1 GeV.

Finally, the best constraints on mH are obtained when all data are used in the fit. The resultsof this fit are shown in column 5 of Table 42 and in Figure 22. In Figures 24 and 25 the sensitivityof the LEP and SLD measurements to the Higgs mass is shown. As can be seen, the most sensitivemeasurements are the asymmetries. A reduced uncertainty for the value of α(m2

Z) would thereforeresult in an improved constraint on mH, as shown in Figures 21 and 26.

In Figure 26 the observed value of ∆χ2 ≡ χ2 − χ2min as a function of mH is plotted for the fit

including all data. The solid curve is the result using ZFITTER, and corresponds to the last columnof Table 42. The shaded band represents the uncertainty due to uncalculated higher-order corrections,as estimated by ZFITTER and TOPAZ0. The 95% confidence level upper limit on mH (taking theband into account) is 165 GeV. The lower limit on mH of approximately 113 GeV obtained fromdirect searches [208] has not been used in this limit determination. Also shown is the result (dashedcurve) obtained when using α(5)(m2

Z) of Reference 198. The fit results in log(mH/GeV) = 1.94+0.22−0.24

corresponding to mH = 88+60−37 GeV and an upper limit on mH of approximately 206 GeV.

- 1 - - 2 - - 3 - - 4 - - 5 -

LEP including all data except all data except all data except all dataLEP-II mW mW and mt mW mt

mt [GeV] 179+13−10 169+10

−8 173.0+4.7−4.5 174.4+9.3

−6.7 174.3+4.4−4.1

mH [GeV] 135+262−83 56+75

−27 74+68−37 61+94

−32 60+52−29

log(mH/GeV) 2.13+0.47−0.41 1.75+0.37

−0.29 1.87+0.28−0.30 1.78+0.41

−0.32 1.78+0.27−0.28

αs(m2Z) 0.120 ± 0.003 0.119 ± 0.003 0.119 ± 0.003 0.118 ± 0.003 0.118 ± 0.003

χ2/d.o.f. 13/9 19/12 20/13 21/14 21/15

sin2θlepteff 0.23167 0.23147 0.23147 0.23140 0.23140

±0.00020 ±0.00017 ±0.00017 ±0.00016 ±0.00016

1−m2W/m

2Z 0.2229 0.2231 0.2229 0.2226 0.2226

±0.0006 ±0.0007 ±0.0005 ±0.0005 ±0.0004

mW [GeV] 80.383 ± 0.029 80.374 ± 0.034 80.386 ± 0.025 80.402 ± 0.025 80.402 ± 0.020

Table 42: Results of the fits to LEP data alone, to all data except the direct determinations of mt andmW (pp collider and LEP-II), to all data except direct mW determinations, and to all data. As thesensitivity to mH is logarithmic, both mH as well as log(mH/GeV) are quoted. The bottom part of thetable lists derived results for sin2θlept

eff , 1 −m2W/m

2Z and mW. See text for a discussion of theoretical

errors not included in the errors above.

71

13 Prospects for the Future

Most of the measurements from data taken at or near the Z resonance, both at LEP as well as atSLC, that are presented in this report are either final or are being finalised. The major improvementswill therefore take place in the high energy data. With more than 600 pb−1 per experiment, LEP-IIwill lead to substantially improved measurements of certain electroweak parameters. As a result, themeasurements of mW are likely to match the error obtained via the radiative corrections of the Zdata, providing a further important test of the Standard Model. In the measurement of the WWγand WWZ triple-gauge-boson couplings the increase in LEP-II statistics, together with the increasedsensitivity at higher beam energies, will lead to an improvement in the current precision.

14 Conclusions

The combination of the many precise electroweak results yields stringent constraints on the StandardModel. All measurements agree with the predictions. In addition, the results are sensitive to theHiggs mass.

The experiments wish to stress that this report reflects a preliminary status at the time of the2000 summer conferences. A definitive statement on these results must wait for publication by eachcollaboration.

Acknowledgements

We would like to thank the CERN accelerator divisions for the efficient operation of the LEP accel-erator, the precise information on the absolute energy scale and their close cooperation with the fourexperiments. The SLD collaboration would like to thank the SLAC accelerator department for theefficient operation of the SLC accelerator. We would also like to thank members of the CDF, DØand NuTeV Collaborations for making results available to us in advance of the conferences and foruseful discussions concerning their combination. Finally, the results of the section on Standard Modelconstraints would not have been possible without the close collaboration of many theorists.

72

0.231

0.2315

0.232

0.2325

0.233

83.6 83.8 84 84.2

∆α

Preliminary 68% CL

Γl [MeV]

sin2 θle

pt

eff

mt= 174.3 ± 5.1 GeVmH= 113...1000 GeV

mt

mH

Figure 21: LEP-I+SLD measurements of sin2θlepteff (Table 40) and Γ`` (Table 4) and the Standard Model

prediction. The point shows the predictions if among the electroweak radiative corrections only thephoton vacuum polarisation is included. The corresponding arrow shows variation of this predictionif α(m2

Z) is changed by one standard deviation. This variation gives an additional uncertainty to theStandard Model prediction shown in the figure.

73

140

160

180

200

10 102

103

mH [GeV]

mt

[GeV

]

Excluded Preliminary

LEP Data, GF, αAll Data

68% CL

Figure 22: The 68% confidence level contours in mt and mH for the fits to LEP data only (dashedcurve) and to all data including the CDF/DØ mt measurement (solid curve). The vertical band showsthe 95% CL exclusion limit on mH from the direct search.

74

80.2

80.3

80.4

80.5

80.6

130 150 170 190 210

mH [GeV]113 300 1000

mt [GeV]

mW

[G

eV]

Preliminary

68% CL

LEP1, SLD, νN Data

LEP2, pp− Data

Figure 23: The comparison of the indirect measurements of mW and mt (LEP-I+SLD+νN data) (solidcontour) and the direct measurements (pp colliders and LEP-II data) (dashed contour). In both casesthe 68% CL contours are plotted. Also shown is the Standard Model relationship for the masses as afunction of the Higgs mass.

75

10 2

10 3

2.49 2.5

ΓZ [GeV]

mH [G

eV]

10 2

10 3

0.015 0.02

AFB0,l

mH [G

eV]

10 2

10 3

41.4 41.5 41.6

σ0h [nb]

mH [G

eV]

10 2

10 3

0.095 0.1 0.105

AFB0,b

mH [G

eV]

10 2

10 3

20.7 20.8

Rl

mH [G

eV]

10 2

10 3

0.06 0.07 0.08

AFB0,c

mH [G

eV]

Preliminary

Figure 24: Comparison of LEP-I measurements with the Standard Model prediction as a function ofmH. The measurement with its error is shown as the vertical band. The width of the Standard Modelband is due to the uncertainties in ∆α(5)

had(m2Z), αs(m2

Z) and mt. The total width of the band is thelinear sum of these effects. See Figure 25 for the definition of these uncertainties.

76

10 2

10 3

0.14 0.15

Aτ

mH [G

eV]

10 2

10 3

0.23 0.232 0.234

sin2Θlepteff from <QFB>

mH [G

eV]

10 2

10 3

0.14 0.15

Ae

mH [G

eV]

10 2

10 3

80.2 80.4

mW [GeV]

mH [G

eV]

10 2

10 3

0.14 0.15

Al (SLD)

mH [G

eV]

Preliminary

Measurement

∆αhad= 0.02804 ± 0.00065∆α(5)

αs= 0.119 ± 0.002

mt= 174.3 ± 5.1 GeV

Figure 25: Comparison of LEP-I measurements with the Standard Model prediction as a function ofmH (c.f. Figure 24). Also shown is the comparison of the SLD measurement of A0

LR with the StandardModel.

77

0

2

4

6

10 102

103

mH [GeV]

∆χ2

Excluded Preliminary

∆αhad =∆α(5)

0.02804±0.00065

0.02755±0.00046

theory uncertainty

Figure 26: ∆χ2 = χ2−χ2min vs. mH curve. The line is the result of the fit using all data (last column

of Table 42); the band represents an estimate of the theoretical error due to missing higher ordercorrections. The vertical band shows the 95% CL exclusion limit on mH from the direct search. Thedashed curve is the result obtained using the evaluation of ∆α(5)

had(m2Z) from Reference 198.

78

Appendix

A Heavy-Flavour Fit including Off-Peak Asymmetries

The full 18 parameter fit to the LEP and SLD data gave the following results:

R0b = 0.21652 ± 0.00069

R0c = 0.1702 ± 0.0034

AbbFB(−2) = 0.0572 ± 0.0075

AccFB(−2) = −0.027 ± 0.016

AbbFB(pk) = 0.0973 ± 0.0021

AccFB(pk) = 0.0621 ± 0.0036

AbbFB(+2) = 0.1106 ± 0.0065

AccFB(+2) = 0.131 ± 0.013

Ab = 0.922 ± 0.022Ac = 0.631 ± 0.026

BR(b → `) = 0.1056 ± 0.0019BR(b → c → ¯) = 0.0801 ± 0.0026

BR(c → `) = 0.0984 ± 0.0032χ = 0.1194 ± 0.0043

f(D+) = 0.237 ± 0.016f(Ds) = 0.121 ± 0.025

f(cbaryon) = 0.090 ± 0.022P(c → D∗+)× BR(D∗+ → π+D0) = 0.1631 ± 0.0050

with a χ2/d.o.f. of 52/(98 − 18). The corresponding correlation matrix is given in Table 43. Theenergy for the peak−2, peak and peak+2 results are respectively 89.55 GeV, 91.26 GeV and 92.94 GeV.Note that the asymmetry results shown here are not the pole asymmetries shown in Section 5.2.2.The non-electroweak parameters do not depend on the treatment of the asymmetries.

79

1) 2) 3) 4) 5) 6) 7) 8) 9) 10) 11) 12) 13) 14) 15) 16) 17) 18)

Rb Rc AbbFB Acc

FB AbbFB Acc

FB AbbFB Acc

FB Ab Ac BR BR BR χ f(D+) f(Ds) f(cbar.) PcDst(−2) (−2) (pk) (pk) (+2) (+2) (1) (2) (3)

1) 1.00 −0.13 0.00 −0.01 −0.02 0.01 −0.01 0.00 −0.04 0.02 −0.09 −0.02 −0.02 −0.02 −0.16 −0.04 0.12 0.112) −0.13 1.00 0.01 0.01 0.05 −0.01 0.02 −0.01 0.02 −0.02 0.05 −0.01 −0.32 0.04 −0.13 0.19 0.18 −0.493) 0.00 0.01 1.00 0.14 0.04 0.01 0.02 0.00 0.01 0.00 0.02 −0.03 0.01 0.06 0.00 0.00 0.00 −0.014) −0.01 0.01 0.14 1.00 0.01 0.02 0.00 0.00 0.00 0.00 0.01 −0.01 0.02 0.01 0.00 0.00 0.00 0.015) −0.02 0.05 0.04 0.01 1.00 0.10 0.10 0.00 0.02 0.00 0.01 −0.09 0.02 0.16 0.01 0.03 0.00 −0.036) 0.01 −0.01 0.01 0.02 0.10 1.00 0.00 0.09 0.00 0.01 0.14 −0.17 −0.05 0.16 0.00 0.00 −0.01 0.007) −0.01 0.02 0.02 0.00 0.10 0.00 1.00 0.13 0.01 0.00 −0.01 −0.03 0.01 0.07 0.01 0.01 0.00 −0.018) 0.00 −0.01 0.00 0.00 0.00 0.09 0.13 1.00 0.00 0.00 0.02 −0.04 −0.03 0.02 −0.01 −0.01 0.00 0.009) −0.04 0.02 0.01 0.00 0.02 0.00 0.01 0.00 1.00 0.14 −0.02 0.00 0.05 0.09 −0.01 0.00 0.00 −0.0110) 0.02 −0.02 0.00 0.00 0.00 0.01 0.00 0.00 0.14 1.00 0.02 −0.03 −0.03 0.01 −0.01 0.00 0.01 0.0111) −0.09 0.05 0.02 0.01 0.01 0.14 −0.01 0.02 −0.02 0.02 1.00 −0.33 0.14 0.43 0.04 0.01 −0.02 −0.0212) −0.02 −0.01 −0.03 −0.01 −0.09 −0.17 −0.03 −0.04 0.00 −0.03 −0.33 1.00 −0.03 −0.40 0.01 −0.01 0.00 0.0113) −0.02 −0.32 0.01 0.02 0.02 −0.05 0.01 −0.03 0.05 −0.03 0.14 −0.03 1.00 0.20 0.02 −0.04 −0.04 0.1614) −0.02 0.04 0.06 0.01 0.16 0.16 0.07 0.02 0.09 0.01 0.43 −0.40 0.20 1.00 0.01 0.01 −0.01 −0.0315) −0.16 −0.13 0.00 0.00 0.01 0.00 0.01 −0.01 −0.01 −0.01 0.04 0.01 0.02 0.01 1.00 −0.38 −0.27 0.1116) −0.04 0.19 0.00 0.00 0.03 0.00 0.01 −0.01 0.00 0.00 0.01 −0.01 −0.04 0.01 −0.38 1.00 −0.46 −0.1117) 0.12 0.18 0.00 0.00 0.00 −0.01 0.00 0.00 0.00 0.01 −0.02 0.00 −0.04 −0.01 −0.27 −0.46 1.00 −0.1718) 0.11 −0.49 −0.01 0.01 −0.03 0.00 −0.01 0.00 −0.01 0.01 −0.02 0.01 0.16 −0.03 0.11 −0.11 −0.17 1.00

Table 43: The correlation matrix for the set of the 18 heavy flavour parameters. BR(1), BR(2) and BR(3) denote BR(b → `), BR(b → c → ¯)and BR(c → `) respectively, PcDst denotes P(c → D∗+)× BR(D∗+ → π+D0).

80

The Measurements used in the Heavy Flavour Averages

In the following 20 tables the results used in the combination are listed. In each case an indication ofthe dataset used and the type of analysis is given. Preliminary results are indicated by the symbol “†”.The values of centre-of-mass energy are given where relevant. In each table, the result used as inputto the average procedure is given followed by the statistical error, the correlated and uncorrelatedsystematic errors, the total systematic error, and any dependence on other electroweak parameters.In the case of the asymmetries, the measurement moved to a common energy (89.55 GeV, 91.26 GeVand 92.94 GeV, respectively, for peak−2, peak and peak+2 results) is quoted as corrected asymmetry.

Contributions to the correlated systematic error quoted here are from any sources of error sharedwith one or more other results from different experiments in the same table, and the uncorrelated errorsfrom the remaining sources. In the case of Ac and Ab from SLD the quoted correlated systematicerror has contributions from any source shared with one or more other measurements from LEPexperiments. Constants such as a(x) denote the dependence on the assumed value of xused, which isalso given.

ALEPH DELPHI L3 OPAL SLD92-95 92-95 94-95 92-95 93-98†[24] [25] [26] [27] [28]

Rb 0.2157 0.2163 0.2174 0.2174 0.2167Statistical 0.0009 0.0007 0.0015 0.0011 0.0009Uncorrelated 0.0007 0.0004 0.0015 0.0009 0.0008Correlated 0.0007 0.0004 0.0018 0.0008 0.0005Total Systematic 0.0009 0.0006 0.0023 0.0012 0.0010a(Rc) -0.0033 -0.0041 -0.0376 -0.0122 -0.0057Rused

c 0.1720 0.1720 0.1734 0.1720 0.1710a(BR(c → `)) -0.0133 -0.0067BR(c → `)used 9.80 9.80a(f(D+)) -0.0010 -0.0010 -0.0086 -0.0029 -0.0008f(D+)used 0.2330 0.2330 0.2330 0.2380 0.2370a(f(Ds)) -0.0001 0.0001 -0.0005 -0.0001 -0.0003f(Ds)

used 0.1020 0.1030 0.1030 0.1020 0.1140a(f(Λc)) 0.0002 0.0003 0.0008 0.0003 -0.0003f(Λc)

used 0.0650 0.0630 0.0630 0.0650 0.0730

Table 44: The measurements of Rb. All measurements use a lifetime tag enhanced by other featureslike invariant mass cuts or high pT leptons.

81

ALEPH DELPHI OPAL SLD91-95 91-95 92-95 92-95 92-95 91-94 90-95 93-97†

c-count D meson lepton c-count D meson c-count D meson vtx-mass[33] [29] [29] [31] [31] [34] [32] [28]

Rc 0.1734 0.1679 0.1668 0.1692 0.1610 0.164 0.1760 0.1732Statistical 0.0049 0.0082 0.0062 0.0047 0.0104 0.011 0.0095 0.0041Uncorrelated 0.0057 0.0078 0.0059 0.0050 0.0064 0.012 0.0102 0.0025Correlated 0.0101 0.0026 0.0010 0.0083 0.0060 0.010 0.0062 0.0004Total Systematic 0.0116 0.0082 0.0059 0.0097 0.0088 0.016 0.0120 0.0025a(Rb) -0.0050 -0.0239Rused

b 0.2159 0.2175a(BR(c → `)) -0.1646BR(c → `)used 9.80

Table 45: The measurements of R0c . “c-count” denotes the determination of R0

c from the sum ofproduction rates of weakly decaying charmed hadrons. “D meson” denotes any single/double taganalysis using exclusive and/or inclusive D meson reconstruction.

82

ALEPH DELPHI L3 OPAL90-95 90-95 90-95 91-95 93-95† 92-95 92-95 90-95 91-95 90-95† 90-95lepton lepton lepton jet charge lepton D-meson jet charge lepton jet charge lepton D-meson

[37] [37] [37] [41] [38] [46] [42] [39] [44] [40] [47]√s (GeV) 88.380 89.380 90.210 89.430 89.433 89.434 89.550 89.500 89.440 89.490 89.490

AbbFB(−2) -3.53 5.47 9.11 7.46 5.90 5.65 6.80 6.14 4.10 3.56 -9.30

AbbFB(−2)Corrected 5.87 7.75 6.18 5.93 6.80 6.26 4.36 3.70 -9.16

Statistical 1.90 1.78 2.20 7.59 1.80 2.93 2.10 1.73 10.80Uncorrelated 0.39 0.19 0.08 0.91 0.12 0.37 0.25 0.16 2.51Correlated 0.70 0.15 0.08 0.08 0.01 0.19 0.02 0.04 1.41Total Systematic 0.80 0.24 0.12 0.91 0.13 0.41 0.25 0.16 2.88a(Rb) -0.3069 -0.2430 -1.1543 -0.1962 -1.4467 -0.7300 -0.1000Rused

b 0.2192 0.2155 0.2164 0.2158 0.2170 0.2150 0.2155a(Rc) 0.0362 1.4800 1.0444 0.3200 0.3612 0.0700 0.1000Rused

c 0.1710 0.1726 0.1671 0.1720 0.1734 0.1730 0.1720a(Acc

FB(−2)) -0.2244 -0.2501 -0.1000 -0.3156Acc

FB(−2)used -2.34 -2.70 -2.50 -2.81a(BR(b → `)) -0.2486 -1.0154 -1.0290 0.3406BR(b → `)used 11.34 10.56 10.50 10.90a(BR(b → c → ¯)) -0.1074 -0.1424 -0.1440 -0.5298BR(b → c → ¯)used 7.86 8.07 8.00 8.30a(BR(c → `)) -0.0474 0.7224 0.5096 0.1960BR(c → `)used 9.80 9.90 9.80 9.80a(χ) 5.259 1.3054χused 0.12460 0.11770a(f(D+)) 0.5083 0.0949f(D+)used 0.2210 0.2330a(f(Ds)) 0.1742 0.0035f(Ds)

used 0.1120 0.1020a(f(Λc)) -0.0191 -0.0225f(Λc)

used 0.0840 0.0630

Table 46: The measurements of AbbFB(−2). All numbers are given in %.

83

ALEPH DELPHI OPAL91-95 93-95† 92-95 90-95† 90-95

D-meson lepton D-meson lepton D-meson[45] [38] [46] [40] [47]√

s (GeV) 89.370 89.433 89.434 89.490 89.490Acc

FB(−2) -1.10 1.11 -5.04 -6.92 3.90

AccFB(−2)Corrected -0.02 1.81 -4.35 -6.56 4.26

Statistical 4.30 3.60 3.69 2.44 5.10Uncorrelated 1.00 0.53 0.40 0.38 0.80Correlated 0.09 0.16 0.09 0.23 0.30Total Systematic 1.00 0.55 0.41 0.44 0.86a(Rb) -0.2886 -3.4000Rused

b 0.2164 0.2155a(Rc) 1.0096 3.2000Rused

c 0.1671 0.1720a(Abb

FB(−2)) -1.3365Abb

FB(−2)used 6.13a(BR(b → `)) -1.0966 -1.7031BR(b → `)used 10.56 10.90a(BR(b → c → ¯)) 1.1156 -1.4128BR(b → c → ¯)used 8.07 8.30a(BR(c → `)) 1.0703 3.3320BR(c → `)used 9.90 9.80a(χ) -0.0856χused 0.11770a(f(D+)) -0.3868f(D+)used 0.2210a(f(Ds)) -0.1742f(Ds)

used 0.1120a(f(Λc)) -0.0878f(Λc)

used 0.0840

Table 47: The measurements of AccFB(−2). All numbers are given in %.

84

ALEPH DELPHI L3 OPAL91-95† 91-95 91-92 93-95† 92-95 92-95 91-95 90-95 91-95 90-95† 90-95lepton jet charge lepton lepton D-meson jet charge jet charge lepton jet charge lepton D-meson

[37] [41] [38] [38] [46] [42] [43] [39] [44] [40] [47]√s (GeV) 91.210 91.250 91.270 91.223 91.235 91.260 91.240 91.260 91.210 91.240 91.240

AbbFB(pk) 9.71 10.40 10.89 9.86 7.59 9.83 9.31 9.85 10.06 9.14 8.90

AbbFB(pk)Corrected 9.81 10.42 10.87 9.93 7.63 9.83 9.35 9.85 10.15 9.18 8.94

Statistical 0.40 0.40 1.30 0.64 1.97 0.47 1.01 0.67 0.52 0.44 2.70Uncorrelated 0.16 0.23 0.33 0.15 0.77 0.14 0.51 0.27 0.41 0.14 2.15Correlated 0.12 0.22 0.27 0.14 0.07 0.04 0.21 0.14 0.20 0.15 0.45Total Systematic 0.20 0.32 0.43 0.20 0.77 0.14 0.55 0.31 0.46 0.20 2.20a(Rb) -0.9545 -0.2430 -2.8933 -2.0201 -0.1962 -9.1622 -2.1700 -7.6300 -0.7000Rused

b 0.2172 0.2155 0.2170 0.2164 0.2158 0.2170 0.2170 0.2150 0.2155a(Rc) 0.6450 1.4900 1.0993 1.1488 0.8400 1.0831 1.3005 0.4600 0.6000Rused

c 0.1720 0.1726 0.1710 0.1671 0.1720 0.1733 0.1734 0.1730 0.1720a(Acc

FB(pk)) 0.6345 1.1603 0.9262 0.6870Acc

FB(pk)used 6.85 6.91 7.41 6.19a(BR(b → `)) -1.8480 -3.8824 -2.0308 -2.0160 -0.3406BR(b → `)used 10.78 11.00 10.56 10.50 10.90a(BR(b → c → ¯)) 0.4233 0.4740 -0.3798 -0.1280 -0.3532BR(b → c → ¯)used 8.14 7.90 8.07 8.00 8.30a(BR(c → `)) 0.5096 0.7840 1.0703 1.5288 0.5880BR(c → `)used 9.80 9.80 9.90 9.80 9.80a(χ) 2.9904 3.4467 1.6692χused 0.12460 0.12100 0.11770a(f(D+)) 0.0442 0.2761f(D+)used 0.2210 0.2330a(f(Ds)) -0.0788 0.0106f(Ds)

used 0.1120 0.1020a(f(Λc)) -0.0115 -0.0495f(Λc)

used 0.0840 0.0630

Table 48: The measurements of AbbFB(pk). All numbers are given in %.

85

ALEPH DELPHI L3 OPAL91-95† 91-95 91-92 93-95† 92-95 90-95 90-95† 90-95lepton D-meson lepton lepton D-meson lepton lepton D-meson

[37] [45] [38] [38] [46] [39] [40] [47]√s (GeV) 91.210 91.220 91.270 91.223 91.235 91.240 91.240 91.240

AccFB(pk) 5.69 6.20 8.05 6.28 6.58 7.94 5.97 6.60

AccFB(pk)Corrected 5.94 6.39 8.00 6.46 6.70 8.04 6.07 6.70

Statistical 0.53 0.90 2.26 1.00 0.97 3.70 0.59 1.20Uncorrelated 0.24 0.23 1.25 0.53 0.25 2.40 0.37 0.49Correlated 0.36 0.17 0.49 0.27 0.04 0.49 0.32 0.24Total Systematic 0.44 0.28 1.35 0.60 0.25 2.45 0.49 0.54a(Rb) 1.4318 2.8933 -2.3087 4.3200 4.1000Rused

b 0.2172 0.2170 0.2164 0.2160 0.2155a(Rc) -2.9383 -6.4736 5.4307 -6.7600 -3.8000Rused

c 0.1720 0.1710 0.1671 0.1690 0.1720a(Abb

FB(pk)) -2.1333 6.4274Abb

FB(pk)used 9.79 8.84a(BR(b → `)) 1.8993 4.8529 -2.7618 3.5007 5.1094BR(b → `)used 10.78 11.00 10.56 10.50 10.90a(BR(b → c → ¯)) -1.0745 -3.9500 2.2786 -3.2917 -1.7660BR(b → c → ¯)used 8.14 7.90 8.07 7.90 8.30a(BR(c → `)) -3.2732 -7.2520 4.8965 -6.5327 -3.9200BR(c → `)used 9.80 9.80 9.90 9.80 9.80a(χ) 0.0453 0.3852χused 0.12460 0.11770a(f(D+)) -0.0221f(D+)used 0.2210a(f(Ds)) 0.0788f(Ds)

used 0.1120a(f(Λc)) 0.0115f(Λc)

used 0.0840

Table 49: The measurements of AccFB(pk). All numbers are given in %.

86

ALEPH DELPHI L3 OPAL90-95 90-95 90-95 91-95 93-95† 92-95 92-95 90-95 91-95 90-95† 90-95lepton lepton lepton jet charge lepton D-meson jet charge lepton jet charge lepton D-meson

[37] [37] [37] [41] [38] [46] [42] [39] [44] [40] [47]√s (GeV) 92.050 92.940 93.900 92.970 92.990 92.990 92.940 93.100 92.910 92.950 92.950

AbbFB(+2) 3.93 10.60 9.03 9.24 10.10 8.78 12.30 13.78 14.60 10.75 -3.40

AbbFB(+2)Corrected 10.03 9.21 10.05 8.73 12.30 13.62 14.63 10.74 -3.41

Statistical 1.51 1.79 1.80 6.37 1.60 2.40 1.70 1.43 9.00Uncorrelated 0.14 0.45 0.14 0.97 0.25 0.34 0.64 0.25 2.03Correlated 0.24 0.26 0.16 0.13 0.05 0.20 0.34 0.28 1.74Total Systematic 0.28 0.52 0.21 0.98 0.26 0.40 0.73 0.37 2.68a(Rb) -1.964 -0.2430 -2.8859 -0.1962 -3.3756 -12.9000 -0.8000Rused

b 0.2192 0.2155 0.2164 0.2158 0.2170 0.2150 0.2155a(Rc) 1.575 1.4900 1.3577 1.2000 1.9869 0.6900 0.8000Rused

c 0.1710 0.1726 0.1671 0.1720 0.1734 0.1730 0.1720a(Acc

FB(+2)) 1.081 1.2018 0.5206 1.3287Acc

FB(+2)used 12.51 12.96 12.39 12.08a(BR(b → `)) -1.762 -2.3557 -2.0790 -1.3625BR(b → `)used 11.34 10.56 10.50 10.90a(BR(b → c → ¯)) -0.2478 -0.7595 -1.1200 0.7064BR(b → c → ¯)used 7.86 8.07 8.00 8.30a(BR(c → `)) 1.524 1.0703 1.9796 0.7840BR(c → `)used 9.80 9.90 9.80 9.80a(χ) 6.584 1.6050χused 0.12460 0.11770a(f(D+)) 0.3978 0.4229f(D+)used 0.2210 0.2330a(f(Ds)) -0.0788 0.0211f(Ds)

used 0.1120 0.1020a(f(Λc)) 0.0573 -0.0855f(Λc)

used 0.0840 0.0630

Table 50: The measurements of AbbFB(+2). All numbers are given in %.

87

ALEPH DELPHI OPAL91-95 93-95† 92-95 90-95† 90-95

D-meson lepton D-meson lepton D-meson[45] [38] [46] [40] [47]√

s (GeV) 92.960 92.990 92.990 92.950 92.950Acc

FB(+2) 10.94 10.50 11.78 15.65 16.70

AccFB(+2)Corrected 10.89 10.37 11.65 15.62 16.67

Statistical 3.30 2.90 3.20 2.02 4.10Uncorrelated 0.79 0.41 0.52 0.57 0.92Correlated 0.18 0.29 0.07 0.62 0.51Total Systematic 0.81 0.50 0.52 0.84 1.05a(Rb) -4.0402 9.6000Rused

b 0.2164 0.2155a(Rc) 7.5891 -8.9000Rused

c 0.1671 0.1720a(Abb

FB(+2)) -2.6333Abb

FB(+2)used 12.08a(BR(b → `)) -3.2492 9.5375BR(b → `)used 10.56 10.90a(BR(b → c → ¯)) 1.5191 -1.5894BR(b → c → ¯)used 8.07 8.30a(BR(c → `)) 8.1341 -9.2120BR(c → `)used 9.90 9.80a(χ) -0.2140χused 0.11770a(f(D+)) -0.2984f(D+)used 0.2210a(f(Ds)) 0.0539f(Ds)

used 0.1120a(f(Λc)) 0.0764f(Λc)

used 0.0840

Table 51: The measurements of AccFB(+2). All numbers are given in %.

88

SLD93-98† 93-98† 94-98† 97-98†lepton jet charge K± multi

[48] [51] [52] [53]√s (GeV) 91.280 91.280 91.280 91.280

Ab 0.922 0.882 0.960 0.926Statistical 0.029 0.020 0.040 0.019Uncorrelated 0.019 0.029 0.056 0.027Correlated 0.008 0.001 0.002 0.001Total Systematic 0.021 0.029 0.056 0.027a(Rb) -0.0542Rused

b 0.2168a(Rc) 0.0424Rused

c 0.1697a(Ac) 0.0449 0.0134 -0.0112 0.0133Ac

used 0.667 0.670 0.666 0.667a(BR(b → `)) -0.2160BR(b → `)used 10.80a(BR(b → c → ¯)) 0.0888BR(b → c → ¯)used 8.05a(BR(c → `)) 0.0479BR(c → `)used 9.83a(χ) 0.3052χused 0.11990

Table 52: The measurements of Ab.

89

SLD93-98† 93-97† 93-97†lepton D-meson K+vertex

[49] [50] [54]√s (GeV) 91.280 91.280 91.280

Ac 0.567 0.688 0.603Statistical 0.051 0.035 0.028Uncorrelated 0.056 0.020 0.023Correlated 0.018 0.003 0.001Total Systematic 0.059 0.021 0.023a(Rb) 0.2173Rused

b 0.2173a(Rc) -0.4089Rused

c 0.1730a(Ab) 0.2151 -0.0673 -0.0306Ab

used 0.935 0.935 0.900a(BR(b → `)) 0.2328BR(b → `)used 11.06a(BR(b → c → ¯)) -0.1178BR(b → c → ¯)used 8.02a(BR(c → `)) -0.4077BR(c → `)used 9.80a(χ) 0.1138χused 0.12170a(f(D+)) -0.0140f(D+)used 0.2300a(f(Ds)) -0.0028f(Ds)

used 0.1150a(f(Λc)) 0.0005f(Λc)

used 0.0740

Table 53: The measurements of Ac.

90

ALEPH DELPHI L3 OPAL91-95† 94-95† 92 94-95† 92-95multi multi lepton multi multi[55] [56] [57] [26] [58]

BR(b → `) 11.55 10.65 10.68 10.21 10.83Statistical 0.09 0.07 0.11 0.13 0.10Uncorrelated 0.17 0.23 0.36 0.20 0.20Correlated 0.22 0.42 0.22 0.31 0.21Total Systematic 0.27 0.48 0.42 0.36 0.29a(Rb) -9.2571 -0.1808Rused

b 0.2160 0.2169a(Rc) 1.4450 0.4867Rused

c 0.1734 0.1770a(BR(b → c → ¯)) -1.1700 0.1618BR(b → c → ¯)used 9.00 8.09a(BR(c → `)) -0.3078 -0.1960 -2.5480 0.9212BR(c → `)used 9.85 9.80 9.80 9.80a(χ) 0.7683χused 0.1178a(f(D+)) 0.5523 0.1445f(D+)used 0.2330 0.2380a(f(Ds)) 0.0213 0.0055f(Ds)

used 0.1030 0.1020a(f(Λc)) -0.0427 -0.0157f(Λc)

used 0.0630 0.0650

Table 54: The measurements of BR(b → `). All numbers are given in %.

91

ALEPH DELPHI OPAL91-95† 94-95† 92-95multi multi multi[55] [56] [58]

BR(b → c → ¯) 8.04 7.88 8.40Statistical 0.14 0.13 0.16Uncorrelated 0.20 0.26 0.19Correlated 0.14 0.36 0.34Total Systematic 0.25 0.45 0.39a(Rb) -0.1808Rused

b 0.2169a(Rc) 0.8916 0.3761Rused

c 0.1694 0.1770a(BR(c → `)) 0.3078 -0.1960BR(c → `)used 9.85 9.80a(χ) -1.2804χused 0.11780a(f(D+)) 0.1190f(D+)used 0.2380a(f(Ds)) 0.0028f(Ds)

used 0.1020a(f(Λc)) -0.0110f(Λc)

used 0.0660

Table 55: The measurements of BR(b → c → ¯). All numbers are given in %.

DELPHI OPAL92-95 90-95

D+lepton D+lepton[30] [59]

BR(c → `) 9.59 9.60Statistical 0.42 0.60Uncorrelated 0.24 0.49Correlated 0.14 0.43Total Systematic 0.27 0.65a(BR(b → `)) -0.5600 -1.4335BR(b → `)used 11.20 10.99a(BR(b → c → ¯)) -0.4100 -0.7800BR(b → c → ¯)used 8.20 7.80

Table 56: The measurements of BR(c → `). All numbers are given in %.

92

ALEPH DELPHI L3 OPAL90-95 94-95† 90-95 90-95†multi multi lepton lepton[37] [56] [39] [40]

χ 0.12461 0.12700 0.11920 0.11390Statistical 0.00515 0.01300 0.00680 0.00540Uncorrelated 0.00252 0.00566 0.00214 0.00306Correlated 0.00397 0.00554 0.00252 0.00324Total Systematic 0.00470 0.00792 0.00330 0.00446a(Rb) 0.0341 0.0000Rused

b 0.2192 0.2170a(Rc) 0.0009 0.0004Rused

c 0.1710 0.1734a(BR(b → `)) 0.0524 0.0550 0.0170BR(b → `)used 11.34 10.50 10.90a(BR(b → c → ¯)) -0.0440 -0.0466 -0.0318BR(b → c → ¯)used 7.86 8.00 8.30a(BR(c → `)) 0.0035 -0.0020 0.0006 0.0039BR(c → `)used 9.80 9.80 9.80 9.80

Table 57: The measurements of χ.


D-meson D-meson[30] [32]

P(c → D∗+) × BR(D∗+ → π+D0) 0.1740 0.1513Statistical 0.0100 0.0096Uncorrelated 0.0040 0.0088Correlated 0.0007 0.0011Total Systematic 0.0041 0.0089a(Rb) 0.0293Rused

b 0.2166a(Rc) -0.0158Rused

c 0.1735

Table 58: The measurements of P(c → D∗+) × BR(D∗+ → π+D0).

93

ALEPH DELPHI OPAL91-95 92-95 91-94

D meson D meson D meson[33] [31] [34]

RcfD+ 0.0406 0.0384 0.0390Statistical 0.0013 0.0013 0.0050Uncorrelated 0.0014 0.0015 0.0042Correlated 0.0032 0.0025 0.0031Total Systematic 0.0035 0.0030 0.0052a(f(D+)) 0.0008f(D+)used 0.2210a(f(Ds)) -0.0002f(Ds)

used 0.1120a(f(Λc)) 0.0000f(Λc)

used 0.0840

Table 59: The measurements of RcfD+.



RcfDs 0.0207 0.0213 0.0160Statistical 0.0033 0.0017 0.0042Uncorrelated 0.0011 0.0010 0.0016Correlated 0.0053 0.0054 0.0043Total Systematic 0.0054 0.0055 0.0046a(f(D+)) 0.0007f(D+)used 0.2210a(f(Ds)) -0.0009f(Ds)

used 0.1120a(f(Λc)) -0.0001f(Λc)

used 0.0840

Table 60: The measurements of RcfDs .

94



RcfΛc 0.0157 0.0169 0.0091Statistical 0.0016 0.0035 0.0050Uncorrelated 0.0005 0.0016 0.0015Correlated 0.0044 0.0045 0.0035Total Systematic 0.0045 0.0048 0.0038a(f(D+)) 0.0002f(D+)used 0.2210a(f(Ds)) -0.0001f(Ds)

used 0.1120a(f(Λc)) -0.0002f(Λc)

used 0.0840

Table 61: The measurements of RcfΛc .



RcfD0 0.0964 0.0926 0.0997Statistical 0.0029 0.0026 0.0070Uncorrelated 0.0040 0.0038 0.0057Correlated 0.0045 0.0023 0.0041Total Systematic 0.0060 0.0044 0.0070a(f(D+)) 0.0020f(D+)used 0.2210a(f(Ds)) -0.0004f(Ds)used 0.1120a(f(Λc)) -0.0004f(Λc)

used 0.0840

Table 62: The measurements of RcfD0.

95


D meson D-meson[31] [32]

RcP(c → D∗+)× BR(D∗+ → π+D0) 0.0282 0.0266Statistical 0.0007 0.0005Uncorrelated 0.0010 0.0010Correlated 0.0007 0.0009Total Systematic 0.0012 0.0014a(f(D+)) 0.0006f(D+)used 0.2210a(f(Ds)) -0.0001f(Ds)

used 0.1120a(f(Λc)) -0.0004f(Λc)

used 0.0840

Table 63: The measurements of RcP(c → D∗+)× BR(D∗+ → π+D0).

96

B The Measurements used in the Electroweak Gauge Boson Cou-

plings

In the following, results from individual experiments used in the electroweak gauge boson couplingsaverages are summarised.

Charged Triple Gauge Boson Couplings

Results from each experiment are shown in Tables 64, 65 and 66 for the single, two and three-parameteranalyses respectively. Uncertainties include both statistical and systematic effects.

Parameter ALEPH [150] DELPHI [151] L3 [152] OPAL [153]

gZ5 +0.13+0.21

−0.21 — −0.04+0.24−0.25 —

∆gz1 +0.008+0.040

−0.039 −0.028+0.046−0.045 −0.075+0.058

−0.054 −0.046+0.043−0.041

∆κγ +0.04+0.09−0.09 +0.05+0.13

−0.12 −0.03+0.12−0.11 −0.10+0.11

−0.10

λγ −0.012+0.041−0.039 +0.056+0.057

−0.056 −0.081+0.065−0.058 −0.108+0.040

−0.038

Table 64: The measured central values and one standard deviation uncertainties (∆ − ∆ logL =0.5) obtained by the four LEP experiments. In each case the parameter listed is varied while theremaining ones are fixed to their Standard Model value. Both statistical and systematic uncertaintiesare included.

Parameter ALEPH [150] DELPHI [151] L3 [152] OPAL [153]

∆gz1 −0.001+0.043

−0.045 −0.048+0.049−0.046 −0.088+0.095

−0.064 −0.017+0.049−0.061

∆κγ +0.02+0.09−0.09 +0.14+0.11

−0.11 +0.04+0.13−0.17 −0.11+0.17

−0.10

∆gz1 +0.019+0.052

−0.051 −0.103+0.063−0.058 −0.02+0.10

−0.16 +0.075+0.059−0.059

λγ −0.026+0.053−0.055 +0.111+0.069

−0.072 −0.06+0.19−0.10 −0.158+0.054

−0.051

λγ −0.019+0.044−0.044 +0.032+0.060

−0.058 −0.088+0.093−0.067 −0.103+0.044

−0.044

∆κγ +0.03+0.09−0.09 +0.07+0.11

−0.10 +0.03+0.12−0.14 −0.03+0.14

−0.11

Table 65: The measured central values and one standard deviation uncertainties (∆ −∆ logL = 0.5)obtained by the four LEP experiments. In each case the two parameters listed are varied while theremaining one is fixed to its Standard Model value. Both statistical and systematic uncertainties areincluded.

97

Parameter ALEPH [150] L3 [152] OPAL [153]

∆gz1 +0.018+0.043

−0.043 −0.02+0.11−0.18 +0.081+0.054

−0.058

∆κγ +0.013+0.071−0.071 −0.02+0.14

−0.15 −0.07+0.11−0.09

λγ −0.028+0.044−0.047 −0.07+0.21

−0.11 −0.149+0.055−0.053

Table 66: The measured central values and one standard deviation uncertainties (∆ −∆ logL = 0.5)obtained by ALEPH, L3 and OPAL. All three parameters listed are varied. Both statistical andsystematic uncertainties are included.

98

Neutral Triple Gauge Boson Couplings in Zγ Production

Results from each experiment are shown in Tables 67 and 68 for the single and two-parameter analysesrespectively. Uncertainties include both statistical and systematic effects.

Parameter DELPHI [154] L3 [155] OPAL [156]

hγ1 [−0.17, +0.17] [−0.14, +0.03] [−0.11, +0.11]

hγ2 [−0.11, +0.11] [−0.039, +0.079] [−0.077, +0.077]

hγ3 [−0.058, +0.051] [−0.095, −0.001] [−0.17, −0.01]

hγ4 [−0.036, +0.041] [+0.005, +0.072] [+0.01, +0.14]

hZ1 [−0.28, +0.29] [−0.16, +0.04] [−0.19, +0.19]

hZ2 [−0.18, +0.18] [−0.042, +0.093] [−0.13, +0.13]

hZ3 [−0.37, +0.21] [−0.16, +0.11] [−0.27, +0.12]

hZ4 [−0.14, +0.22] [−0.05, +0.11] [−0.08, +0.18]

Table 67: The 95% C.L. intervals (∆−∆ logL = 1.92) measured by DELPHI, L3 and OPAL. In eachcase the parameter listed is varied while the remaining ones are fixed to their Standard Model value.Both statistical and systematic uncertainties are included.

Parameter DELPHI [154] L3 [155]

hγ1 [−0.35, +0.35] [−0.23, +0.10]

hγ2 [−0.23, +0.23] [−0.12, +0.10]

hγ3 [−0.26, +0.38] [−0.20, +0.12]

hγ4 [−0.32, +0.43] [−0.10, +0.12]

hZ1 [−0.58, +0.58] [−0.49, +0.19]

hZ2 [−0.38, +0.38] [−0.31, +0.16]

hZ3 [−0.70, +0.50] [−0.37, +0.40]

hZ4 [−0.41, +0.37] [−0.20, +0.28]

Table 68: The 95% C.L. intervals (∆ −∆ logL = 1.92) measured by DELPHI and L3. In each casethe two parameters listed are varied while the remaining ones are fixed to their Standard Model value.Both statistical and systematic uncertainties are included.

99

Neutral Triple Gauge Boson Couplings in ZZ Production

Results from each experiment are shown in Tables 69 and 70 for the single and two-parameter analysesrespectively. Uncertainties include both statistical and systematic effects.


fγ4 [−0.49, +0.49] [−0.59, +0.58] [−0.45, +0.43]

fZ4 [−0.82, +0.83] [−0.97, +0.99] [−0.74, +0.75]

fγ5 [−1.2, +1.2] [−1.4, +1.4] [−0.9, +0.8]

fZ5 [−1.9, +2.1] [−2.2, +2.5] [−1.0, +0.5]

Table 69: The 95% C.L. intervals (∆−∆ logL = 1.92) measured by DELPHI, L3 and OPAL. In eachcase the parameter listed is varied while the remaining ones are fixed to their Standard Model value.Both statistical and systematic uncertainties are included.


fγ4 [−0.49, +0.49] [−0.58, +0.58] [−0.45, +0.43]

fZ4 [−0.82, +0.83] [−0.97, +0.99] [−0.73, +0.75]

fγ5 [−1.2, +1.2] [−1.4, +1.4] [−0.9, +0.9]

fZ5 [−1.9, +2.1] [−2.2, +2.5] [−1.0, +0.7]

Table 70: The 95% C.L. intervals (∆−∆ logL = 1.92) measured by DELPHI, L3 and OPAL. In eachcase the two parameters listed are varied while the remaining ones are fixed to their Standard Modelvalue. Both statistical and systematic uncertainties are included.

100

Quartic Gauge Boson Couplings

Results from each experiment are shown in Table 71, where the uncertainties include both statisticaland systematic effects.

Parameter ALEPH [160] L3 [161] OPAL [162]

[GeV−2]

a0/Λ2 [−0.043, +0.042] [−0.036, +0.035] [−0.065, +0.065]

ac/Λ2 [−0.11, +0.11] [−0.08, +0.11] [−0.13, +0.17]

an/Λ2 — [−0.45, +0.42] [−0.61, +0.57]

a0/Λ2 — [−0.006, +0.006] [−0.006, +0.008]

ac/Λ2 — [−0.006, +0.010] [−0.008, +0.012]

Table 71: The 95% C.L. intervals (∆ −∆ logL = 1.92) measured by ALEPH, L3 and OPAL. In eachcase the parameter listed is varied while the remaining ones are fixed to their Standard Model value.Both statistical and systematic uncertainties are included. Top: WWγ and ννγγ. Bottom: Zγγ.

101

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